{
 "cells": [
  {
   "cell_type": "markdown",
   "metadata": {
    "colab_type": "text",
    "id": "view-in-github"
   },
   "source": [
    "<a href=\"https://colab.research.google.com/github/jermwatt/machine_learning_refined/blob/main/notes/16_Linear_algebra/16_2_Vectors.ipynb\" target=\"_parent\"><img src=\"https://colab.research.google.com/assets/colab-badge.svg\" alt=\"Open In Colab\"/></a>"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "id": "N-lz7Owne4Ly"
   },
   "source": [
    "## Chapter 16: Elements of linear algebra"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "This notebook contains interactive content from an early draft of the university textbook <a href=\"https://github.com/neonwatty/machine-learning-refined/tree/main\">\n",
    "Machine Learning Refined (2nd edition) </a>.\n",
    "\n",
    "The final draft significantly expands on this content and is available for <a href=\"https://github.com/neonwatty/machine-learning-refined/tree/main/chapter_pdfs\"> download as a PDF here</a>."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "id": "-NHyFrXOe4Lz"
   },
   "source": [
    "# C.2 Vectors and Vector Operations"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "id": "re52GJfhe4L0"
   },
   "source": [
    "In this Section we review the concept of a elementary arithmetic performed with vectors, sometimes referred to as 'arrays'.  "
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 1,
   "metadata": {
    "id": "MYYb20kce4L0"
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Requirement already satisfied: github-clone in /Users/jeremywatt/Desktop/machine-learning-refined/venv/lib/python3.10/site-packages (1.2.0)\n",
      "Requirement already satisfied: requests>=2.20.0 in /Users/jeremywatt/Desktop/machine-learning-refined/venv/lib/python3.10/site-packages (from github-clone) (2.32.3)\n",
      "Requirement already satisfied: docopt>=0.6.2 in /Users/jeremywatt/Desktop/machine-learning-refined/venv/lib/python3.10/site-packages (from github-clone) (0.6.2)\n",
      "Requirement already satisfied: charset-normalizer<4,>=2 in /Users/jeremywatt/Desktop/machine-learning-refined/venv/lib/python3.10/site-packages (from requests>=2.20.0->github-clone) (3.4.0)\n",
      "Requirement already satisfied: idna<4,>=2.5 in /Users/jeremywatt/Desktop/machine-learning-refined/venv/lib/python3.10/site-packages (from requests>=2.20.0->github-clone) (3.10)\n",
      "Requirement already satisfied: urllib3<3,>=1.21.1 in /Users/jeremywatt/Desktop/machine-learning-refined/venv/lib/python3.10/site-packages (from requests>=2.20.0->github-clone) (2.2.3)\n",
      "Requirement already satisfied: certifi>=2017.4.17 in /Users/jeremywatt/Desktop/machine-learning-refined/venv/lib/python3.10/site-packages (from requests>=2.20.0->github-clone) (2024.12.14)\n",
      "\n",
      "\u001b[1m[\u001b[0m\u001b[34;49mnotice\u001b[0m\u001b[1;39;49m]\u001b[0m\u001b[39;49m A new release of pip is available: \u001b[0m\u001b[31;49m24.2\u001b[0m\u001b[39;49m -> \u001b[0m\u001b[32;49m24.3.1\u001b[0m\n",
      "\u001b[1m[\u001b[0m\u001b[34;49mnotice\u001b[0m\u001b[1;39;49m]\u001b[0m\u001b[39;49m To update, run: \u001b[0m\u001b[32;49mpip install --upgrade pip\u001b[0m\n",
      "chapter_16_library already cloned!\n",
      "chapter_16_videos already cloned!\n"
     ]
    }
   ],
   "source": [
    "# install github clone - allows for easy cloning of subdirectories\n",
    "!pip install github-clone\n",
    "from pathlib import Path \n",
    "\n",
    "# clone library subdirectory\n",
    "if not Path('chapter_16_library').is_dir():\n",
    "    !ghclone https://github.com/neonwatty/machine-learning-refined-notes-assets/tree/main/notes/16_Linear_algebra/chapter_16_library\n",
    "else:\n",
    "    print('chapter_16_library already cloned!')\n",
    "\n",
    "# clone videos\n",
    "if not Path('chapter_16_videos').is_dir():\n",
    "    !ghclone https://github.com/neonwatty/machine-learning-refined-notes-assets/tree/main/notes/16_Linear_algebra/chapter_16_videos\n",
    "else:\n",
    "    print('chapter_16_videos already cloned!')\n",
    "        \n",
    "\n",
    "# append path for local library, data, and image import\n",
    "import sys\n",
    "sys.path.append('./chapter_16_library') \n",
    "\n",
    "# backend file\n",
    "from linear_algebra_library import vector_plots, transform_animators\n",
    "\n",
    "# video paths\n",
    "video_path_1 = 'chapter_16_videos/animation_1.mp4'\n",
    "video_path_2 = 'chapter_16_videos/animation_1.mp4'\n",
    "\n",
    "# standard imports\n",
    "from matplotlib.axes._axes import _log as matplotlib_axes_logger\n",
    "matplotlib_axes_logger.setLevel('ERROR')\n",
    "import matplotlib.pyplot as plt\n",
    "from IPython.display import Image, HTML\n",
    "import autograd.numpy as np\n",
    "from base64 import b64encode\n",
    "\n",
    "def show_video(video_path, width = 1000):\n",
    "    video_file = open(video_path, \"r+b\").read()\n",
    "    video_url = f\"data:video/mp4;base64,{b64encode(video_file).decode()}\"\n",
    "    return HTML(f\"\"\"<video width={width} controls><source src=\"{video_url}\"></video>\"\"\")\n",
    "\n",
    "# this is needed to compensate for matplotlib notebook's tendancy to blow up images when plotted inline\n",
    "%matplotlib inline\n",
    "from matplotlib import rcParams\n",
    "rcParams['figure.autolayout'] = True\n",
    "\n",
    "%load_ext autoreload\n",
    "%autoreload 2"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "id": "X5EnuPVOe4L8"
   },
   "source": [
    "##  Vectors and vector operations"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "id": "ghuYsHLse4L9"
   },
   "source": [
    "### Row and column vectors, dimensions"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "id": "S0Im0OZTe4L9"
   },
   "source": [
    "A *vector* is another word for *a listing of numbers*.  For example the following\n",
    "\n",
    "$$\n",
    "[2.1, \\, \\, -5.7, \\, \\, 13]\n",
    "$$\n",
    "\n",
    "is a vector of three elements, also referred to as a vector of length three or a vector of dimension $1\\times3$.  The '1' in the first position tells us that this is a row vector, while the '3' says how many elements the vector has.  In general a vector can be of arbitrary length, and can contain numbers, variables, or both.  For example\n",
    "\n",
    "$$\n",
    "[x_1, \\,\\, x_2, \\,\\, x_3, \\,\\, x_4]\n",
    "$$\n",
    "\n",
    "is a vector of four variables, of dimension $1\\times4$."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "id": "XeiiSR6Oe4L9"
   },
   "source": [
    "When listing the numbers / variables out horizontally we call the vector a *row vector*.  Of course we can also list them just as well vertically, e.g., we could write the first example above as a column\n",
    "\n",
    "\\begin{bmatrix}\n",
    "2.1  \\\\\n",
    "-5.7 \\\\\n",
    "13 \\\\\n",
    "\\end{bmatrix}\n",
    "\n",
    "in which case we refer to this as a *column vector* of length three or a vector of dimension $3\\times 1$.  Notice that the row version of this had dimension $1\\times 3$.  Here the '1' in the second entry tells us that the vector is a column.   "
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "id": "lfQe0ouSe4L9"
   },
   "source": [
    "### Transposing vectors"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "id": "w0oxtddLe4L-"
   },
   "source": [
    "We can swap back and forth between a row and column version of a vector by *transposing* it.  This is an operation performed on a single vector, and simply turns a row vector into an equivalent column vector and vice-versa.  This operation is denoted by a *superscript T* placed just to the right and above a vector.  For example we can transpose a column into a row vector like this\n",
    "\n",
    "$$\n",
    "{\\begin{bmatrix}\n",
    "2.1  \\\\\n",
    "-5.7 \\\\\n",
    "13 \\\\\n",
    "\\end{bmatrix}}^{\\,T}\n",
    "= [2.1, \\, \\, -5.7, \\, \\, 13]\n",
    "$$\n",
    "\n",
    "and likewise a row into a column vector like this \n",
    "\n",
    "$$\n",
    "[2.1, \\, \\, -5.7, \\, \\, 13]^{\\,T} = \n",
    "{\\begin{bmatrix}\n",
    "2.1  \\\\\n",
    "-5.7 \\\\\n",
    "13 \\\\\n",
    "\\end{bmatrix}}\n",
    "$$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "id": "HLihLIM9e4L-"
   },
   "source": [
    "### Vector notation"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "id": "fnYOMkSUe4L-"
   },
   "source": [
    "To discuss vectors more generally we use algebraic notation, typically a *bold lowercase (often English) letter*.  This notation does not denote whether or not the vector is a row or column, or how many elements it contains: such information must be given explicitly.  For example we can denote a vector of numbers\n",
    "\n",
    "$$\n",
    "\\mathbf{x} = [2.1, \\, \\, -5.7, \\, \\, 13]\n",
    "$$\n",
    "\n",
    "Here the fact that $\\mathbf{x}$ represents a row vector with three elements is clear from its definition.  Thus when we say $\\mathbf{x}^T$ it is clear from its definition that\n",
    "\n",
    "$$\n",
    "\\mathbf{x} = {\\begin{bmatrix}\n",
    "2.1  \\\\\n",
    "-5.7 \\\\\n",
    "13 \\\\\n",
    "\\end{bmatrix}}\n",
    "$$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "id": "GpB1hrg-e4L_"
   },
   "source": [
    "We could also define $\\mathbf{x}$ to be the vector of variables like\n",
    "\n",
    "$$\n",
    "\\mathbf{x} = \n",
    "{\\begin{bmatrix}\n",
    "x_1  \\\\\n",
    "x_2 \\\\\n",
    "x_3 \\\\\n",
    "x_4 \\\\\n",
    "\\end{bmatrix}}\n",
    "$$\n",
    "\n",
    "Again nothing about the notation $\\mathbf{x}$ itself tells us whether or not it is a row or column vector, nor how many elements it contains.  This information is given explicitly when we define what the notation means here. "
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "id": "L7xbJ7cOe4L_"
   },
   "source": [
    "### The geometric interpretation of vectors"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "id": "uUxFc_Dqe4L_"
   },
   "source": [
    "Vectors are often interpreted geometrically, and can be drawn when in two or three dimensions.  A single vector is usually drawn as either a point or an arrow stemming from the origin.  In the next cell we illustrate both.  The 'point' version is in the left panel, with the 'arrow' version being in the right.  In both cases we simply plot each coordinate of the vector as a coordinate in the Cartesian plane.  Regardless of whether a vector is a column or row it is drawn the same."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 2,
   "metadata": {
    "id": "4_5vXsv9e4L_",
    "outputId": "707d037d-a7e5-4e89-bb4f-c985dce7cb2b"
   },
   "outputs": [
    {
     "data": {
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",
      "text/plain": [
       "<Figure size 1500x500 with 2 Axes>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "# import numpy, define a vectors\n",
    "import numpy as np\n",
    "vec1 = np.asarray([3,3])\n",
    "plotter = vector_plots.single_plot(vec1)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "id": "7SHaOh17e4MA"
   },
   "source": [
    "### Defining a vector in numpy, transpose operation"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "id": "KJJqfut1e4MA"
   },
   "source": [
    "Vectors are referred to more generally as 'arrays', which is the nomenclature used to constructs a vector in numpy as shown in the following cell."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 3,
   "metadata": {
    "id": "nj5dBB_Ne4MA",
    "outputId": "5dc7ca43-f55b-40c6-fb26-d05fce7d7f97"
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "[ 2.1 -5.7 13. ]\n"
     ]
    }
   ],
   "source": [
    "# import statement for numpy\n",
    "import numpy as np   \n",
    "\n",
    "# construct a vector (a.k.a. an array), and print it out\n",
    "x = np.asarray([2.1,-5.7,13])\n",
    "print (x)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "id": "fL8bvvbje4MB"
   },
   "source": [
    "By default a numpy array is initialized in this way is *dimensionless* - technically speaking neither a row nor a column vector - which you can see by printing its 'shape' which is numpy-speak for dimensions."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 4,
   "metadata": {
    "id": "BrosJA29e4MB",
    "outputId": "3a76af7f-9bfe-4df0-ed19-ce9832bc65ed"
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "(3,)\n"
     ]
    }
   ],
   "source": [
    "# print out the vector's initial shape (or dimensions)\n",
    "print (np.shape(x))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "id": "j4xW96JBe4MB"
   },
   "source": [
    "Thus we must explicitly define whether $\\mathbf{x}$ is a row or column vector.  We can do this by re-defining its shape as shown in the next cell."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 5,
   "metadata": {
    "id": "30fOcSUWe4MB",
    "outputId": "4caf5c91-6646-4be9-ba92-d0be55a6131d"
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "----- x as a row vector ----\n",
      "[[ 2.1 -5.7 13. ]]\n",
      "----- x as a column vector ----\n",
      "[[ 2.1]\n",
      " [-5.7]\n",
      " [13. ]]\n"
     ]
    }
   ],
   "source": [
    "# reshape x to be a row vector and print\n",
    "x.shape = (1,3) \n",
    "print ('----- x as a row vector ----')\n",
    "print (x)\n",
    "\n",
    "# reshape x to be a column vector and print\n",
    "x.shape = (3,1)\n",
    "print ('----- x as a column vector ----')\n",
    "print (x)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "id": "sr0eHLrue4MC"
   },
   "source": [
    "The notation for transposing a vector in numpy looks like\n",
    "\n",
    "                <numpy_array>.T\n",
    "                \n",
    "We illustrate on $\\mathbf{x}$ in the next cell.  Notice that we last set $\\mathbf{x}$ to be a column vector prior to activating the cell below."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 6,
   "metadata": {
    "id": "R-1XIYrOe4MC",
    "outputId": "88a5a07b-db36-48bf-dab3-b7ab269b7d9a"
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "----- the original vector - a column -----\n",
      "[[ 2.1]\n",
      " [-5.7]\n",
      " [13. ]]\n",
      "----- the transpose - now a row vector ----- \n",
      "[[ 2.1 -5.7 13. ]]\n"
     ]
    }
   ],
   "source": [
    "print ('----- the original vector - a column -----')\n",
    "print (x)\n",
    "print ('----- the transpose - now a row vector ----- ')\n",
    "print (x.T)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "id": "l8-KrtFDe4MC"
   },
   "source": [
    "##  Adding and subtracting vectors"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "id": "8re45esFe4MC"
   },
   "source": [
    "We add and subtract two vectors elementwise, with just one catch: in order to add/subtract two vectors they must have the same dimensions.  This means that in order to add/subtract two vectors they must have the same number of elements, and both must be row or column vectors."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "id": "oyV5qVape4MC"
   },
   "source": [
    "For example, to add these two vectors\n",
    "\n",
    "$$\n",
    "\\mathbf{x} = {\\begin{bmatrix}\n",
    "2.1  \\\\\n",
    "-5.7 \\\\\n",
    "13 \\\\\n",
    "\\end{bmatrix}} \\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\n",
    "\\mathbf{y} = {\\begin{bmatrix}\n",
    "4.3  \\\\\n",
    "9.2 \\\\\n",
    "13 \\\\\n",
    "\\end{bmatrix}}\n",
    "$$\n",
    "\n",
    "we add them element-wise\n",
    "\n",
    "$$\n",
    "\\mathbf{x} + \\mathbf{y} = {\\begin{bmatrix}\n",
    "2.1 + 4.3  \\\\\n",
    "-5.7 + 9.2 \\\\\n",
    "13 + 13 \\\\\n",
    "\\end{bmatrix}} = {\\begin{bmatrix}\n",
    "6.4  \\\\\n",
    "3.5 \\\\\n",
    "26 \\\\\n",
    "\\end{bmatrix}}\n",
    "$$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "id": "loTUPYDoe4MD"
   },
   "source": [
    "We likewise subtract these two vectors as\n",
    "\n",
    "$$\n",
    "\\mathbf{x} - \\mathbf{y} = {\\begin{bmatrix}\n",
    "2.1 - 4.3  \\\\\n",
    "-5.7 - 9.2 \\\\\n",
    "13 - 13 \\\\\n",
    "\\end{bmatrix}} = {\\begin{bmatrix}\n",
    "-2.2  \\\\\n",
    "-14.9 \\\\\n",
    "0 \\\\\n",
    "\\end{bmatrix}}\n",
    "$$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "id": "Nlsybpase4MD"
   },
   "source": [
    "We can add / subtract vectors in numpy as shown in the next Python cell."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 7,
   "metadata": {
    "id": "YC4wgO8ee4MD",
    "outputId": "8c6e7cf8-f861-44eb-9ebe-bb72bc77df45"
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "*** x + y ***\n",
      "[[ 6.4]\n",
      " [ 3.5]\n",
      " [26. ]]\n",
      "*** x - y ***\n",
      "[[ -2.2]\n",
      " [-14.9]\n",
      " [  0. ]]\n"
     ]
    }
   ],
   "source": [
    "# define both x and y, make x a row vector and y a column vector\n",
    "x = np.asarray([2.1,-5.7,13])\n",
    "x.shape = (3,1)\n",
    "y = np.asarray([4.3, 9.2, 13])\n",
    "y.shape = (3,1)\n",
    "print ('*** x + y ***')\n",
    "print (x + y)\n",
    "print ('*** x - y ***')\n",
    "print (x - y)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "id": "qEjrlqU6e4MD"
   },
   "source": [
    "More generally to add two Nx1 column vectors \n",
    "\n",
    "$$\n",
    "\\mathbf{x} = {\\begin{bmatrix}\n",
    "x_1 \\\\\n",
    "x_2\\\\\n",
    "\\vdots \\\\\n",
    "x_N \\\\\n",
    "\\end{bmatrix}} \\,\\,\\,\\,\\,\\,\\,\\,\\,  \\mathbf{y} = {\\begin{bmatrix}\n",
    "y_1 \\\\\n",
    "y_2\\\\\n",
    "\\vdots \\\\\n",
    "y_N \\\\\n",
    "\\end{bmatrix}}\n",
    "$$\n",
    "\n",
    "we write \n",
    "\n",
    "$$\n",
    "\\mathbf{x} + \\mathbf{y} = {\\begin{bmatrix}\n",
    "x_1 + y_1 \\\\\n",
    "x_2 + y_2\\\\\n",
    "\\vdots \\\\\n",
    "x_N + y_N \\\\\n",
    "\\end{bmatrix}}\n",
    "$$\n",
    "\n",
    "and likewise for subtraction."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "id": "Seq8lcLpe4MD"
   },
   "source": [
    "### Adding / subtracting two vectors of different dimensions"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "id": "EUE4CE9We4MD"
   },
   "source": [
    "Elementwise addition / subtraction is by far the most common type of addition/ subtraction used in practice with vectors, and is by default what we assume in the future when we say describe addition / subtraction of vectors unless stated otherwise.  "
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "id": "IefgfZkAe4ME"
   },
   "source": [
    "Even if two vectors have the same number of elements, techically speaking we cannot add or subtract them if one is a row vector and the other is a column vector.  However with numpy it is possible to add two vectors of different shapes via numpy's built in [broadcasting](https://docs.scipy.org/doc/numpy-1.10.1/user/basics.broadcasting.html) operations.  \n",
    "\n",
    "For example if $\\mathbf{x}$ was a row vector\n",
    "\n",
    "$$\\mathbf{x} = [2.1, \\, \\, -5.7, \\, \\, 13]$$\n",
    "\n",
    "and $\\mathbf{y}$ was a column vector\n",
    "\n",
    "$$\\mathbf{y} = {\\begin{bmatrix}\n",
    "4.3  \\\\\n",
    "9.2 \\\\\n",
    "13 \\\\\n",
    "\\end{bmatrix}}\n",
    "$$\n",
    "\n",
    "addition/subtraction with $\\mathbf{y}$ would not be defined.  If we try this in numpy we will not throw an error, but return a matrix of values."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 8,
   "metadata": {
    "id": "tqSZdX-me4ME",
    "outputId": "5378f542-947f-4e81-a133-b380107a48ce"
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "*** x + y ***\n",
      "[[ 6.4 -1.4 17.3]\n",
      " [11.3  3.5 22.2]\n",
      " [15.1  7.3 26. ]]\n"
     ]
    }
   ],
   "source": [
    "# turn x into a row vector\n",
    "x.shape = (1,3)\n",
    "\n",
    "# try to add x and y\n",
    "print ('*** x + y ***')\n",
    "print (x + y)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "id": "9pl1l0dae4ME"
   },
   "source": [
    "Examining the matrix closely, you can see that what numpy has done here is make three copies of $\\mathbf{y}$, and added the first element of $\\mathbf{x}$ to each element of the first copy, added the second element of $\\mathbf{x}$ to the second copy, and the third element of $\\mathbf{x}$ to the third copy.  Numpy makes this sort of operation on $\\mathbf{x}$ and $\\mathbf{y}$ more convenient than having to use a for loop."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "id": "nDXLVIU7e4ME"
   },
   "source": [
    "If we try to add / subtract two vectors of different lengths numpy will throw an error.  For example, in the next cell we try to add a vector with three elements to that has only two."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 9,
   "metadata": {
    "id": "D178S0vre4ME",
    "outputId": "cea81d5c-aacd-433f-ab95-37bb5b9eb6eb"
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "*** x + y ***\n"
     ]
    },
    {
     "ename": "ValueError",
     "evalue": "operands could not be broadcast together with shapes (3,1) (2,1) ",
     "output_type": "error",
     "traceback": [
      "\u001b[0;31m---------------------------------------------------------------------------\u001b[0m",
      "\u001b[0;31mValueError\u001b[0m                                Traceback (most recent call last)",
      "Cell \u001b[0;32mIn[9], line 7\u001b[0m\n\u001b[1;32m      5\u001b[0m y\u001b[38;5;241m.\u001b[39mshape \u001b[38;5;241m=\u001b[39m (\u001b[38;5;241m2\u001b[39m,\u001b[38;5;241m1\u001b[39m)\n\u001b[1;32m      6\u001b[0m \u001b[38;5;28mprint\u001b[39m (\u001b[38;5;124m'\u001b[39m\u001b[38;5;124m*** x + y ***\u001b[39m\u001b[38;5;124m'\u001b[39m)\n\u001b[0;32m----> 7\u001b[0m \u001b[38;5;28mprint\u001b[39m (\u001b[43mx\u001b[49m\u001b[43m \u001b[49m\u001b[38;5;241;43m+\u001b[39;49m\u001b[43m \u001b[49m\u001b[43my\u001b[49m)\n",
      "\u001b[0;31mValueError\u001b[0m: operands could not be broadcast together with shapes (3,1) (2,1) "
     ]
    }
   ],
   "source": [
    "# define both x and y, make x a row vector and y a column vector\n",
    "x = np.asarray([2.1,-5.7,13])\n",
    "x.shape = (3,1)\n",
    "y = np.asarray([4.3, 9.2])\n",
    "y.shape = (2,1)\n",
    "print ('*** x + y ***')\n",
    "print (x + y)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "id": "E6hjWkyqe4MF"
   },
   "source": [
    "### The geometric interpretation of element-wise vector addition"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "id": "Tn8gyxsue4MF"
   },
   "source": [
    "Two-dimensions vectors are often represented geometrically by plotting each vector not as a point, but as an arrow stemming from the origin.  From this perspective the addition of two vectors can be seen to be (very nicely) always be equal to the vector representing the far corner of the parallelogram formed by the two vectors in the sum.  This is called the *parallelogram law*, and is illustrated by the Python cell below for any two user-defined input vectors. \n",
    "\n",
    "Here the two input vectors are colored black, with their sum shown in red.  Note the blue dashed lines are merely visual guides helping to outline the  parallelogram underlying the sum."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {
    "id": "pvYMVjYOe4MF",
    "outputId": "1b493107-a2ed-4a58-ac23-2685503f8c1b"
   },
   "outputs": [
    {
     "data": {
      "image/png": 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      "text/plain": [
       "<Figure size 1500x500 with 3 Axes>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "# import numpy, define two vectors, and add to see their sum visually\n",
    "import numpy as np\n",
    "vec1 = np.asarray([1,3])\n",
    "vec2 = np.asarray([-3,2])\n",
    "plotter = vector_plots.vector_add_plot(vec1,vec2)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "id": "YvHdaSmPe4MF"
   },
   "source": [
    "##  Vector multiplication"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "id": "GMf6mXB1e4MG"
   },
   "source": [
    "### Multiplying a vector by a scalar"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "id": "1WGNQiYje4MG"
   },
   "source": [
    "We can multiply any vector by a scalar by treating the multiplication elementwise.  For example if \n",
    "\n",
    "$$\n",
    "\\mathbf{x} = {\\begin{bmatrix}\n",
    "2.1  \\\\\n",
    "-5.7 \\\\\n",
    "13 \\\\\n",
    "\\end{bmatrix}}\n",
    "$$\n",
    "\n",
    "then for any scalar $c$\n",
    "\n",
    "$$\n",
    "c\\times\\mathbf{x} = {\\begin{bmatrix}\n",
    "c\\times 2.1  \\\\\n",
    "c\\times-5.7 \\\\\n",
    "c\\times13 \\\\\n",
    "\\end{bmatrix}}\n",
    "$$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "id": "QBpl_AD6e4MG"
   },
   "source": [
    "And this is how scalar multiplication is done regardless of whether or not the vector is a row or column."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "id": "ZGM4mw39e4MG"
   },
   "source": [
    "Using numpy we can use the standard multiplication operator to perform scalar-vector multiplication, as illustrated in the next cell."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {
    "id": "32QbeCa4e4MG",
    "outputId": "fbf3de51-ce06-4ae4-a4cf-7317536f0fd7"
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "[  4.2 -11.4  26. ]\n"
     ]
    }
   ],
   "source": [
    "# define vector\n",
    "x = np.asarray([2.1,-5.7,13])\n",
    "\n",
    "# multiply by a constant\n",
    "c = 2\n",
    "print (c*x)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "id": "f-yBWMOje4MH"
   },
   "source": [
    "This holds in general for a general $N\\times 1$ vector $\\mathbf{x}$ as well.\n",
    "\n",
    "$$\n",
    "c\\times\\mathbf{x} = {\\begin{bmatrix}\n",
    "c\\times x_1 \\\\\n",
    "c\\times x_2\\\\\n",
    "\\vdots \\\\\n",
    "c\\times x_N \\\\\n",
    "\\end{bmatrix}}\n",
    "$$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "id": "fmRgKZ7Se4MH"
   },
   "source": [
    "### The elementwise product"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "id": "g_BT3WTie4MH"
   },
   "source": [
    "There are a number of ways to multiply two vectors - perhaps the most natural is the *elementwise product*.  This works precisely how it sounds: multiply two vectors of the same dimension element-by-element.  The former piece of this is important: just like addition, technically speaking we need both vectors to have the same dimension in order to make this work. \n",
    "\n",
    "To multiply two vectors element-wise"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "id": "aGVTcI2fe4MH"
   },
   "source": [
    "$$\n",
    "\\mathbf{x} = {\\begin{bmatrix}\n",
    "x_1 \\\\\n",
    "x_2\\\\\n",
    "\\vdots \\\\\n",
    "x_N \\\\\n",
    "\\end{bmatrix}} \\,\\,\\,\\,\\,\\,\\,\\,\\,  \\mathbf{y} = {\\begin{bmatrix}\n",
    "y_1 \\\\\n",
    "y_2\\\\\n",
    "\\vdots \\\\\n",
    "y_N \\\\\n",
    "\\end{bmatrix}}\n",
    "$$\n",
    "\n",
    "we then write \n",
    "\n",
    "$$\n",
    "\\mathbf{x} \\times \\mathbf{y} = {\\begin{bmatrix}\n",
    "x_1 \\times y_1 \\\\\n",
    "x_2 \\times y_2\\\\\n",
    "\\vdots \\\\\n",
    "x_N \\times y_N \\\\\n",
    "\\end{bmatrix}}\n",
    "$$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "id": "91CdkyeXe4MH"
   },
   "source": [
    "In numpy we use the natural multiplication operation '*' to perform elementwise multiplication between two vectors."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {
    "id": "K5o2YjwOe4MH",
    "outputId": "681906d5-cc51-46fc-b37d-dd08b9954b65"
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "[[  9.03]\n",
      " [-52.44]\n",
      " [169.  ]]\n"
     ]
    }
   ],
   "source": [
    "# define vector\n",
    "x = np.asarray([2.1,-5.7,13])\n",
    "x.shape = (3,1)\n",
    "y = np.asarray([4.3, 9.2, 13])\n",
    "y.shape = (3,1)\n",
    "print (x*y)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "id": "tu-8fyrUe4MI"
   },
   "source": [
    "### The inner product"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "id": "rCLgPlPHe4MI"
   },
   "source": [
    "The inner product is another way to multiply two vectors of the same dimension, and is the natural extension of multiplication of two scalar values in that this product produces a scalar output.  Here is how the inner product is defined: to take the inner product of two $N\\times1$ vectors we first multiply them together entry-wise, then add up the result.\n",
    "\n",
    "For two general vectors \n",
    "\n",
    "$$\n",
    "\\mathbf{x} = {\\begin{bmatrix}\n",
    "x_1 \\\\\n",
    "x_2\\\\\n",
    "\\vdots \\\\\n",
    "x_N \\\\\n",
    "\\end{bmatrix}} \\,\\,\\,\\,\\,\\,\\,\\,\\,  \\mathbf{y} = {\\begin{bmatrix}\n",
    "y_1 \\\\\n",
    "y_2\\\\\n",
    "\\vdots \\\\\n",
    "y_N \\\\\n",
    "\\end{bmatrix}}\n",
    "$$\n",
    "\n",
    "the inner product is written as $\\mathbf{x}^T \\mathbf{y}$ and is defined as\n",
    "\n",
    "$$\n",
    "\\mathbf{x}^T \\mathbf{y}= \\text{sum}\\left(\\mathbf{x}\\times \\mathbf{y}\\right)  =x_{1}y_{1}+x_{2}y_{2}+\\cdots x_{N}y_{N} = \\sum_{n=1}^Nx_ny_n\n",
    "$$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "id": "R3Jj0aAie4MI"
   },
   "source": [
    "The inner product is also often referred to as the 'dot' product, and written notationally as "
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "id": "_sEN7SWSe4MI"
   },
   "source": [
    "$$\n",
    "\\mathbf{x}\\odot\\mathbf{y}\n",
    "$$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "id": "3GSfGs38e4MI"
   },
   "source": [
    "In the next cell we use numpy to take the inner product between two vectors.  Notice that we can write this out in at least two ways:\n",
    "\n",
    "1.  By using the transpose notation given above: ``np.dot(x.T,y)``\n",
    "2.  By using the element-wise product then sum notation: ``np.sum(x*y)``"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {
    "id": "1FQmvllqe4MI",
    "outputId": "6b3e2d40-c274-4699-cb20-520c7faec77a"
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "125.59\n",
      "125.59\n"
     ]
    }
   ],
   "source": [
    "# define two column vectors and compute the inner product\n",
    "x = np.asarray([2.1,-5.7,13])\n",
    "x.shape = (3,1)\n",
    "y = np.asarray([4.3, 9.2, 13])\n",
    "y.shape = (3,1)\n",
    "\n",
    "# compute the inner product\n",
    "print (np.dot(x.T,y)[0][0])\n",
    "print(np.sum(x*y))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "id": "-Mp5YNbNe4MJ"
   },
   "source": [
    "### The inner product and the norm (or geometric length) of a vector"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "id": "LyZhgqQEe4MJ"
   },
   "source": [
    "The inner product (also commonly called the *correlation*) is also interesting because it helps define the geometric length of a vector.  Notice the result of multiplying a vector $\\mathbf{x}$ by itself\n",
    "\n",
    "$$\n",
    "\\mathbf{x}^T \\mathbf{x}= \\sum_{n=1}^Nx_nx_n = \\sum_{n=1}^Nx_n^2\n",
    "$$\n",
    "\n",
    "We square each element and sum the result.  Visualizing a vector - as we do in the next Python cell for a two-dimensional example - we can square this formula with a common elementary formula for the vector's length.  In the left panel below we show the 'point' view of a two-dimensional vector, and in the right panel the corresponding 'arrow' version.  Visual guides - drawn in dashed blue - are shown in each panel."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {
    "id": "0SfVM6Pae4MJ",
    "outputId": "8126f888-bddc-4e88-d11f-18ab2ce99b4d"
   },
   "outputs": [
    {
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",
      "text/plain": [
       "<Figure size 1500x500 with 2 Axes>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "# import numpy, define a vectors\n",
    "import numpy as np\n",
    "vec1 = np.asarray([3,3])\n",
    "plotter = vector_plots.single_plot(vec1,guides = True)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "id": "ialyHkQKe4MJ"
   },
   "source": [
    "If we treated this vector - say as drawn in the right panel - as the hypotonus of a right triangle, how would we calculate its length?  Using the *Pythagorean theorem* we would square the length of each side - i.e., the length of each blue guide - sum the result, and take the square root of this.  Or - denoting our two-dimensional vector more generally as \n",
    "\n",
    "$$\n",
    "\\mathbf{x} = {\\begin{bmatrix}\n",
    "x_1 \\\\\n",
    "x_2\\\\\n",
    "\\end{bmatrix}}\n",
    "$$\n",
    "\n",
    "then this computation becomes\n",
    "\n",
    "$$\n",
    "\\text{length of hypotonus (a.k.a. length of vector)} = \\sqrt{x_1^2 + x_2^2}\n",
    "$$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "id": "z2Uj9sb2e4MJ"
   },
   "source": [
    "In terms of the inner product we can write this length computation equivalently as \n",
    "\n",
    "$$\n",
    "\\sqrt{x_1^2 + x_2^2} = \\sqrt{\\mathbf{x}^T\\mathbf{x}}\n",
    "$$\n",
    "\n",
    "Therefore we can express the length of a vector in terms of the inner product with itself - and this generalizes to vectors of any length."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "id": "LZlJysQ4e4MJ"
   },
   "source": [
    "Thus the notation denoting the length of a vector $\\mathbf{x}$ being $\\lVert \\mathbf{x} \\rVert _2$, we can always compute the length as\n",
    "\n",
    "$$\n",
    "\\lVert \\mathbf{x} \\rVert _2 = \\sqrt{\\mathbf{x}^T\\mathbf{x}}\n",
    "$$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "id": "gG6oUXO4e4MK"
   },
   "source": [
    "### The geometric interpretation of the inner product"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "id": "v9nEi5Hce4MK"
   },
   "source": [
    "The following beautiful formula - referred to as the *inner product rule* - holds for any two vectors $\\mathbf{x}$ and $\\mathbf{y}$  \n",
    "\n",
    "$$\n",
    "\\mathbf{x}^T\\mathbf{y} = \\lVert \\mathbf{x} \\rVert_2 \\lVert \\mathbf{y} \\rVert_2 \\text{cos}(\\theta)\n",
    "$$\n",
    "\n",
    "This rule - which can be formally proven using the *law of cosines* from trigonometry - is perhaps best intuited after a slight rearrangement of its terms, as follows.\n",
    "\n",
    "$$\n",
    "\\left(\\frac{\\mathbf{x}}{ \\lVert \\mathbf{x} \\rVert_2}\\right)^T \\left(\\frac{\\mathbf{y}}{ \\lVert \\mathbf{y} \\rVert_2} \\right)= \\text{cos}(\\theta)\n",
    "$$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "id": "9qckQ8kue4MK"
   },
   "source": [
    "Here each input vector is normalized, i.e,. has length equal to one, and the formula can be interpreted as a smooth measurement of the angle $\\theta$ between these two vectors lying on the unit circle. \n",
    "\n",
    "Notice that because cosine lies between $-1$ and $+1$ so too does this measurement.  When the two vectors point in the exact same direction the value takes on $+1$, when they point in completely opposite directions $-1$.  When the two vectors are *perpendicular* to each other, their inner product is equal to zero.  \n",
    "\n",
    "The two extremes $\\pm 1$ are easy to verify - for example if $\\mathbf{x} = \\mathbf{y}$ then the two vectors overlap completely, pointing in the same direction and\n",
    "\n",
    "$$\n",
    "\\left(\\frac{\\mathbf{x}}{ \\lVert \\mathbf{x} \\rVert_2}\\right)^T \\left(\\frac{\\mathbf{x}}{ \\lVert \\mathbf{x} \\rVert_2} \\right)=  \\left(\\frac{\\mathbf{x}^T\\mathbf{x}}{ \\lVert \\mathbf{x} \\rVert_2^2} \\right)=\\left(\\frac{\\lVert \\mathbf{x} \\rVert_2^2}{ \\lVert \\mathbf{x} \\rVert_2^2} \\right) = +1\n",
    "$$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "id": "o8gqfzWoe4MK"
   },
   "source": [
    "In the next Python cell we animate the value of the inner product for two unit-length vectors using a slider widget.  The first $\\mathbf{x}_1 = \\begin{bmatrix} 1 \\\\ 0 \\end{bmatrix}$ is fixed, while the second is free to vary around the circle using the slider.  As the free vector rotates around the circle the corresponding inner product between it and the fixed vector is shown in the right panel.  Note how when the free vector is perpendicular to the fixed vector, the inner product is zero.  When they are pointed in completely opposite directions the inner product is $-1$, and when perfectly aligned $+1$. "
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {
    "id": "WlezVM7we4MK"
   },
   "outputs": [],
   "source": [
    "# # illustrate the full range of the inner product for against the vector [1,0] - uncomment to re-render animation\n",
    "# start = 0.45  # where to start off on the unit circle\n",
    "# transform_animators.inner_product_visualizer(video_path_1,num_frames = 200,start = start,fps=40)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {
    "id": "xUVMfg4Ve4MK",
    "outputId": "0f476666-5e9e-45ae-e860-c00e1e2d8720"
   },
   "outputs": [
    {
     "data": {
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      ],
      "text/plain": [
       "<IPython.core.display.HTML object>"
      ]
     },
     "execution_count": 2,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "show_video(video_path_1)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "id": "s0wuUQjfe4ML"
   },
   "source": [
    "### The  outer product"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "id": "0AeZGrcle4ML"
   },
   "source": [
    "The **outer product** is another way to define multiplication between two vectors.  With two column vectors $\\mathbf{x}$ and $\\mathbf{y}$\n",
    "\n",
    "$$\n",
    "\\mathbf{x}=\\begin{bmatrix}\n",
    "x_{1}\\\\\n",
    "x_{2}\\\\\n",
    "\\vdots\\\\\n",
    "x_{N}\n",
    "\\end{bmatrix},\\,\\,\\,\\,\\,\\,\\,\\mathbf{y}=\\begin{bmatrix}\n",
    "y_{1}\\\\\n",
    "y_{2}\\\\\n",
    "\\vdots\\\\\n",
    "y_{M}\n",
    "\\end{bmatrix}\n",
    "$$\n",
    "\n",
    "their *outer product* is written as $\\mathbf{x}\\mathbf{y}^T$ and is defined as\n",
    "\n",
    "$\\mathbf{x}\\mathbf{y}^{T}=\\begin{bmatrix}\n",
    "x_{1}\\\\\n",
    "x_{2}\\\\\n",
    "\\vdots\\\\\n",
    "x_{N}\n",
    "\\end{bmatrix}\\begin{bmatrix}\n",
    "y_{1}\\\\\n",
    "y_{2}\\\\\n",
    "\\vdots\\\\\n",
    "y_{M}\n",
    "\\end{bmatrix}^{T}=\\begin{bmatrix}\n",
    "x_{1}y_{1} & x_{1}y_{2} & \\cdots & x_{1}y_{M}\\\\\n",
    "x_{2}y_{1} & x_{2}y_{2} & \\cdots & x_{2}y_{M}\\\\\n",
    "\\vdots & \\vdots & \\ddots & \\vdots\\\\\n",
    "x_{N}y_{1} & x_{N}y_{2} & \\cdots & x_{N}y_{M}\n",
    "\\end{bmatrix}$\n",
    "\n",
    "This is an $N\\times M$ *matrix* - which can be thought of as a collection of $M$ column vectors stacked side-by-side (or - likewise as a collection of $N$ row vectors stacked one on top of each other). We'll soon discuss matrices further."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "id": "pldhkgvWe4ML"
   },
   "source": [
    "In Python we can compute this outer product using the notation ``outer``."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {
    "id": "MvutWOlbe4ML",
    "outputId": "40a5f437-75d4-4286-e7cf-7abf429fbf09"
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "[[  9.03  19.32]\n",
      " [-24.51 -52.44]\n",
      " [ 55.9  119.6 ]]\n"
     ]
    }
   ],
   "source": [
    "# define vector\n",
    "x = np.asarray([2.1,-5.7,13])\n",
    "x.shape = (3,1)\n",
    "y = np.asarray([4.3, 9.2])\n",
    "y.shape = (2,1)\n",
    "print (np.outer(x,y.T))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "id": "djavDn0je4ML"
   },
   "source": [
    "## Linear combination of vectors"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "id": "qJbiHFSBe4ML"
   },
   "source": [
    "A linear combination is an operation that generalizes simple addition of two vectors by combining addition and scalar multiplication. Given two vectors $\\mathbf{x}_{1}$ and $\\mathbf{x}_{2}$ of the same dimension, their linear combination is formed by multiplying each with a scalar first and then adding up the result, as in    \n",
    "\n",
    "$$\\alpha_{1}\\mathbf{x}_{1}+\\alpha_{2}\\mathbf{x}_{2}$$\n",
    "\n",
    "where $\\alpha_{1}$ and $\\alpha_{2}$ are real numbers. Notice that for a given set of values for $\\alpha_{1}$ and $\\alpha_{2}$, the linear combination is a vector itself with the same dimension as $\\mathbf{x}_{1}$ and $\\mathbf{x}_{2}$. For instance with $\\alpha_{1}=2$ and $\\alpha_{2}=4$, the linear combination is given by\n",
    "\n",
    "$$2\\mathbf{x}_{1}+4\\mathbf{x}_{2}$$\n",
    "\n",
    "In the Python cell below we plot this linear combination for two-dimensional vectors $\\mathbf{x}_{1} = [2, \\, \\, 1]$ and $\\mathbf{x}_{2} = [-1, \\, \\, 2]$."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {
    "id": "ZCszCe68e4MM",
    "outputId": "85c5d8ca-c08f-448e-908c-766ab9135097"
   },
   "outputs": [
    {
     "data": {
      "image/png": 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      "text/plain": [
       "<Figure size 1500x500 with 3 Axes>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "# This code cell will not be shown in the HTML version of this notebook\n",
    "vec1 = np.asarray([2,1])\n",
    "vec2 = np.asarray([-1,2])\n",
    "alpha1 = 2\n",
    "alpha2 = 5\n",
    "\n",
    "plotter = vector_plots.vector_linear_combination_plot(vec1, vec2, alpha1, alpha2)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "id": "UZPUdfWZe4MM"
   },
   "source": [
    "Let's try a different set of values for $\\alpha_{1}$ and $\\alpha_{2}$."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {
    "id": "G-1a8Egwe4MM",
    "outputId": "ab1200cd-ac21-4502-eb8c-f116dd99a4cc"
   },
   "outputs": [
    {
     "data": {
      "image/png": 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AADgsOTk2PfhgglasiNIRR/g1enSFJOn114ssTobfYsvKUuoppwSN16Snq/DttynPcUjuvrtcklRVZSg62rQ4DQAADYOV6AAAAAAOSm6uTa+/Hq2rrkrSmjWRSkgI6OSTa7V0ab4WL87XpZdWWx0RvyNu4sT9Fuj58+er8J13KNBxyCIi6v679tokzZwZbXUcAAAaBCvRAQAAAPyuX36JUIcOfs2a5dJf/5qgfv1q9Ic/VOu447xyuaQ//7nS6og4ALa8PKX27h00Xtu7twrmz5dsrLNCw3jggVJddlmyunf36uSTvVbHAQDgsFCiAwAAANivTZvsev99lzIzncrPj9DatbnKyPDoggs8bNPQAsU+/bTiH388aLxg1izV7mdPdOBwHHusT48+WqLRo5O0YkWe4uN5zAAAtFyU6AAAAAAkSbW10iefRCk726Y//rFa77/vUlWVoUceKdWpp9bKzl8PLZJRVKS0nj2Dxr1duyp/+fK6vTeARjB0qEcdO/op0AEALR6/BgMAAABhzu+Xxo1rpaVLnUpL8+/Z0/yvfy23OBkOV8zUqUqYMCFovHDmTNWcc44FiRBuTjrJq/XrHVq82Kl77+UxBQDQMlGiAwAAAGGmoMCmxYudcrudOuOMGo0ZU6l+/Wo0bly5unTxWx0PDcAoK1Najx5B4/7UVOV+8onkcFiQCuHqyCP9mjfPpS5d/Pq//6uyOg4AAAeNU2MAAACAMLBzZ4Sqq6WNGx069dQ2WrTIqYwMz55V55ddVk2BHiKiZ87cb4FeNH26cj//nAIdTS4pKaDp04v08MPx+uIL5h8AoOVhJToAAAAQonbsiNDcudHKzHTql18i9OabherVy6uvvspVbCx7FIcao6pKaUcfHTQeiItTzpdfSk6nBamAOscf79Njj5WovNywOgoAAAeNlegAAABAiPD5pDVrIvXUU7GSpK+/jlRRkU3331+mjRtz1Lu3VxERokAPQa7Zs/dboBdPmaKcrVsp0NEs/OEPHvXrV6tFi5zyeq1OAwDAgWMlOgAAANCCmaZkGNKjj8Zp5swYJSf7lZHhUW2tNGxYtYYNq7Y6IhqTx6PU44+XrTr465y9fbvM6GgLQgH188WVqkEAACAASURBVPul6dNj9PHHkXr44TKr4wAAcEAo0QEAAIAWpqjI0NKlTmVmOuV0mnruuRKlp9fq8sur1bWrz+p4aCLODz5Q0g03BI2XPP64qq680oJEwO+LiJBeeqlYGRnJ6tnTqyuu4Ik+AEDzR4kOAAAAtAC//BIhu91UQoKp9PS2OumkWg0a5NHAgR5JUr9+NRYnRJOprVXbvn0VUVAQdCl761aZcXEWhAIOXOvWAU2bVqy7707QJZdUy04zAQBo5vhRBQAAADRT2dk2vfVWtNxup374wa7HHivVJZdUa8OGHLlcVqeDFaKWLVPrESOCxksfekiVo0ZZkAg4NCec4NUHH9Q9EVRWZig+nrMaAADNFyU6AAAA0Ez4/dL69ZFyu5265ZZyFRXZlJ0dobvuKld6es2esyEp0MOQz6c2/fvLvnNn0KXsTZtkJiY2fSbgMEVESK++Gq3333fpzTcL5XBYnQgAgP2zWR0AAAAACGeBQN3b11+PVu/ebXX77a1ks0l+v6HjjvPpn/8s1Xnn/a9AR/iJXL1a7Y44IqhAL7vnHu3KyqJAR4t25ZVVCgSkhx+OtzoKAAD1YiU6AAAA0MRKSgwtX+6U2+3U1q12rVqVr5NPrtU77xTqmGN8MgyrE6JZCASUPHiwIjdtCrqU8+WXCqSkWBAKaFgOx+6DRlN05pk1GjiQ8x0AAM0PJToAAADQBLKzbSopsalHD5+uvLK1YmNNZWR49MADZTIMqUcPn9UR0Yw41q1TysUXB42X33qryu++24JEQONJSQlo1qwCtW/vtzoKAAD7RYkOAAAANJLyckMzZsTI7XZq2za7xo6tUI8eFZo3r4C9f7F/pqnWl1+uqLVrgy7lrF+vQFqaBaGAxnfkkX5VVhoaPTpRTz5ZotatA1ZHAgBgD/ZEBwAAABpIICB9/rlDEyfGad06h+x2Uz/+GKFbb63Qxo25uv32CkmiQMd+Ob7+Wu06dAgq0CtGjdKurCwKdIS8mBhTaWl+jR2bKB8vzgEANCOU6AAAAMBh2F30rFwZpT592urGGxNVU2MoMdGUyyU98USpBg3yyOUyrQ2K5ss0lXTNNUrJyAi6lLt2rcoeesiCUIA1JkwoVU2NoYkTOWgUANB8sJ0LAAAAcJDKyw2tWBElt9upFSucWrkyT8ce69XrrxfquOM4GBQHzr51q9qcd17QeOXw4Sp98kkLEgHWioyUpk4t0rx5LpmmeDwFADQLlOgAAADAAcjLs2n7drvS02v18MPx+u47uzIyPLrnnnK1bVu3d2+bNuzhiwNkmmp1882Kfu+9oEu5q1bJ37WrBaGA5qFt24BuuKFSn3/ukNNp6rjj2NsFAGAtSnQAAACgHqYpvfRSjBYtcmnzZrsuuMCj9PRaPfZYqWxsjIhDFPH992p71llB49UXXaTiF15g6S3wX1u3OjR5cqwWLsxXUhJbYgEArEOJDgAAAPyXaUobNzrkdjt1xBE+DR9ercpKm0aPrlD//jWKiakrcSjQcagS7rpLMW+8ETSet3SpfD16WJAIaL6uuqpKX33l0I03JmnmzELZaTAAABbhRxAAAADCmtdb9/aXXyJ0+eXJMk0pI8Ojbt3qtg8YP77cwnQIFRE//6y2p50WNO4591wVvfYaq8+Bejz8cKmuvrq1tmxxqGdPr9VxAABhihIdAAAAYaeqytDKlVFatMipZcuceu65Yp11Vo2mTSvSCSd46TPRoOInTFDs1KlB4/kLF8p7wgkWJAJajqgo6e23C2UYUm6ubc8ZFAAANCVeiAoAAICwUFho07x5TknS7NkuvfhirI491qv338/XOefUyG6XTjyRAh0Nx5aTo3bt2wcV6DWnnKJdP/9MgQ4cIMOQsrJsOvfcNtqyhbWAAICmR4kOAACAkPbOOy717y/17dtGs2ZFq7LS0J/+VKX58ws0dmyljjrKb3VEhKC4xx9Xap8+QeMFc+eqcO5cNtYHDlL79gHddVeZRo1KUnExz3YCAJoWT+ECAAAgZJimtHmzXZmZLlVWGvr738skSTfeKL38cq5iY9kGAI3LVlio1F69gsa9xx6rfLdbioiwIBUQGkaMqNLGjQ79/e8Jmjy5xOo4AIAwQokOAACAFs3nkzweQ06nqXPOaaOKCkMDB3p04YXVkqQrrqhWWlorZWebMk2LwyKkxT7/vOIfeSRovODNN1Xbr58FiYDQYhjSI4+UqriYV3IAAJoWJToAAABanOpq6aOPouR2u7RkSZTGjKnUTTdVaPr0InXt6mOnDDQpo6REaccdFzTuO+II5X34oWTnzy6goTidUlpaQH//e7xOPbVWF1zgsToSACAM8OcFAAAAWoTiYkPvvutScbGhHTscevbZOHXp4tOcOYW66aYKSVK3bhToaFrRM2bst0AvnDFDeR9/TIEONJIhQzy6665W+vZbvscAAI2PnzYAAABo1lasiNILL8Tq888dOuWUWp14Yq169vRqwYICq6MhjBkVFUo75pigcX/r1spdv16KjLQgFRA+TjutVnfcUXfQ6Acf5Cshgf26AACNhxIdAAAAzYZpStu22eV2O7V+faRefbVILpep4cOrNHWqR61aUZLAeq633lLi+PFB40UvvijPRRdZkAgITyNHVskwxCuQAACNjhIdAAAAlvL7peJim5KTAxo5MklffOHQwIEejRhRKdOsW20INAdGdbVSu3WTEQj8atx0OpW9aZPkclmUDAhPhlFXpOfn27RwoVN//GO11ZEAACGKEh0AAABNzuvdfTCoU4sXO3XmmTWaMqVEEyeWKDU1oIgIqxMCv+acN09JN94YNF781FOqvuIKCxIB2K221tDEifFKSDCVkcFBowCAhkeJDgAAgCZRVmZo+XKnjj7aqyOP9Onpp+N0zjkevflmobp390mS2rcP/M69AE2spkapvXvLVloadCl72zaZMTEWhAKwt/bt/XrhhWJdf32SunYtUNeuPqsjAQBCDDuHAQAAoFFt2mTXlVcmqXfvVM2cGa2SEptcLmn+/AKNG1ehHj18MgyrUwLBnJmZanfkkUEFesnEidqVlUWBDjQjZ5xRq3HjyrVkSZTVUQAAIYiV6AAAAGhQO3ZEKDPTJbfbqRdfLFKrVqYuvrhaU6aUKCmJleZoAbxetTnjDNl37Qq6lP3NNzITEiwIBeD3/PnPlZKkn3+OUPv2fg4cBQA0GH6kAAAA4LAEAtJPP9VtYj5xYpyGDk3R9u123XRThVq3DqhDB7+uuKKaAh0tQtSqVWrXuXNQgV56//11q88p0IFmb9y4VnrmmVirYwAAQggr0QEAAHBIPvwwSosW1R0MmpQU0OLF+RozpkJ33VUuO79loqXx+5Vy/vlybNsWdCnn668VSEqyIBSAQ/H00yUaMiRZxx/v1YABNVbHAQCEAFaiAwAA4IBUVBhasMCpxYvr9pudPdulpKSAXn21UIsX58swpKQkkwIdLU7kJ5+oXadOQQV62R13aFdWFgU60MJ06ODX888X6847W6migkM3AACHjz9xAAAA8Jvy820aP76V1qyJVK9eXl1zTZUk6ZlnSixOBhymQEDJQ4cqcsOGoEs5GzYo0LatBaEANIQzz6xVZma+YmNNmaY4wBoAcFhYiQ4AAIBf2bkzQi++GKNhw1orM9OpxMSAhgyp1qef5mnu3EJdfHG11RGBw+b44gu169gxqECvGDu2bvU5BTrQ4rVtG9Dy5VEaMyZRAY7lAAAcBkp0AACAMGea0jff1L1Acc4clwYPTtHmzQ6NGlWpM8+skd0uDR9ereRkGgiEANOUBg5U8gUXBF3K/ewzld13nwWhADSW006r1Xff2TVlCgeNAgAOHdu5AAAAhKnPP3do3jyX3G6n/H5Dixfna/Dgal14YbUiI61OBzQ8++bNajNwYNB45YgRKn30UQsSAWhs0dGmXnmlSBdcUHfQ6LnnctAoAODgUaIDAACEiaoqQ6tWRSkrK0J//nOlPv00Si6XqZdeKtYJJ3hl4zWKCFWmqcQbbpBr4cKgS7lr1sjfuXPTZwLQZDp18mvq1GKlpPCKKgDAoaFEBwAACGGmWfd29OhELV8epe7dfXv2NL/xxgoLkwFNw75jh9r07x80XnXZZSp5+mlOGwTCxOmn18rnkyZPjtV111UqJsa0OhIAoAWhRAcAAAgxP/8cocxMp9xup/r2rdXdd5frssuq9NBDpUpNZRUewoRpqtW4cYqeNSv42ubNKk1M/N+zTADCQkSEtGmTQ+PGtdJLLxXzHBoA4IDxol0AAIAWbvfBoGVlhrZutev881O0fn2krr66UmPH1q02HziwhgIdYSPixx/VrkOHoAK9etAgZWdlSccea1EyAFYyDOlf/yrRd9/Z9fzzHDQKADhwrEQHAABoobZutevtt6OVmelUVZWhadOK1KePV199lSOn0+p0gDUS/vY3xcyYETSel5kp3/HHy2DpKRDWYmJMTZ9epEceiZffX7c6HQCA30OJDgAA0EJ4PNJHH0Vp9eooPfBAmX7+OUI2m/TMM8U66STvniKAAh3hyJaVpdRTTgkar0lPV+Hbb7P3OYA9Onf26+WXi1VWZqi83FD79rxSCwDw2yjRAQAAmrFAQLLZpH/8I16vvRatrl19GjTIo9paacCAGg0YUGN1RMBycRMnKu6554LG8+fPl7dPHwsSAWgJ5sxxaebMGM2fX6DoaM5IAADUjxIdAACgmcnOtikz06nMTKfsdun114s0eHC1rr22gtVywF5seXlK7d07aLy2d28VzJ9f9wwUANTjmmuqtGZNlMaPb6Xnn+egUQBA/fitEgAAwGKmKW3bZtePP0aotlYaPDhFa9ZE6dJLq/Xss8WSpD59vBTowF5in356vwV6wbvvqmDBAgp0AL/LMKSnnirR1q12LVkSZXUcAEAzxkp0AGFl7ty5evPNNzVkyBCNHDnS6jgAwlxWlk0zZsTI7XapuNjQhAlluvTSaq1fnys7v6UB+2UUFSmtZ8+gce/RRyt/2TJOCQRwUGJjTc2ZU6BWrUwOGgUA1IvlGQDCxo4dO7R06VIdccQRVkcBEKZqaqQVK6J0990JysqyqbbWUG2toccfL9GXX+bq0kurJYkCHahHzNSp+y3QC994Q/krV9J+ATgkiYmmyssNnX9+in76iccRAEAw/kQDEBY8Ho8mT56s0aNHa86cOVbHARBGamulyEjptdei9cgj8erUya+MDI8iI6X27f2aMKHM6ohAs2eUlSmtR4+gcX9amnLXrpUcDgtSAQgl8fGm/vCHao0alaT58wusjgMAaGYo0QGEhWnTpql3797q1avX75boXq9XXq93z/uGYcjlcu35N5qX3V8TvjbYbe85YdW8yMurOxjU7Y7S1q0OffZZnvr1q9U55xSoUyf/3mktyRdueJxo2aJff10Jd98dNF40fbpqBg8+6O8i5gP2xZzAbrfdVqmvvrLrL39xacEC5gQA4H8o0QGEvDVr1uiHH37Qo48+ekC3nzt3rt59990973fp0kWTJk1SSkpKY0VEA0hNTbU6ApqJ7OxsSVJycrLS0tKa7PNu3y4VFkqnnSb96U9STEzd24suklJS0tShQ5NFQT14nGhhKiul2Njg8fh4KTdXSU7nYd098wH7Yk6Et2+//VYTJkzQp59+pPPPf1DSKOYEAGAPSnQAIa2goEAzZszQ3/72N0VGRh7QxwwbNkwXXnjhnvd3r0DJz8+Xz+drlJw4dIZhKDU1VTk5OTJN0+o4aAYKCgr2vN1dqDeWigpDU6bEyu2OUl5ehK6/vlJHHFGh11+Xdi9e8/mkRo6B38HjRMvjmj1brW65JWi8eMoUeS65RCouPuT7Zj5gX8yJ8JSfn6/FixfL7XZr4sSJMgxD3bp10y233KIuXbpIUtjMCbvdzoIhAPgdlOgAQtr333+v0tJS3XPPPXvGAoGAtmzZIrfbrf/85z+y2X59xrLD4ZCjnr1Vw+GX6JbKNE2+PpD0v+/TxpgTXq/0ySeRcrtdGjjQo/T0Gnk80kMPleq002rlcEi7PyXTsfnhcaIF8HiUevzxslVXB13K3r5dZnR0g31zMR+wL+ZE6CsqKlJSUpKmTJmip556SmeeeaYGDx6shIQExcXF6YYbbpDUuL9LAABaJkp0ACGtZ8+eeuKJJ3419sILL6hdu3YaOnRoUIEOAPuqrjbkcpn68MNIjRmTpNRUvwYN8qhTJ5/sdumBBzgYFGgIzg8+UNJ/C6y9lTz+uKquvNKCRABCwddff61FixYpMzNTFRUVWrt2rS6//HKNHDlSsfvbMgoAgP2gRAcQ0lwulzp16vSrsaioKMXFxQWNA8BuhYU2LVkSJbfbpdWrI7ViRb569fJqwYJ8HXmk//fvAMCBq61V2759FfHfrZj2lr11q8y4OAtCAWipvF6vPv30U3m9Xp1zzjl67rnn1Lp1az3wwAM6/fTTZbPZ1LZtW6tjAgBaGEp0AAAAST/+GKEtWxzKyPDohRdi9e23dmVkePTPf5aoTZuAJKlVKwp0oCFFLVum1iNGBI2XPvSQKkeNsiARgJaqpKREDzzwgJYuXao2bdro2muvlSS9+OKLFicDAIQCSnQAYefBBx+0OgKAZuSZZ2L1/vsu/fJLhIYM8Sgjw6P77mOLFqBR+Xxq07+/7Dt3Bl3K3rRJZmJi02cC0KIUFRVpyZIlcrvdGj58uM4//3wde+yxuvXWW3XUUUdZHQ8AEGIo0QEAQNjw+aTPPouU2+1Uhw5+3XBDpaKjTd13X5lOP71GUVFWJwRCX+Tq1Ur+4x+Dxsv++ldV3HyzBYkAtBRZWVlq166dlixZorFjx+r000/XoEGD1KdPH0VERGj06NFWRwQAhChKdAAAENKqq+veFhVFaNCgZCUmmho8uFpnnFEjSbr++koL0wFhJBBQSkaGHJs3B13K+eorBZKTLQgFoLnbtm2bFixYILfbrR9//FErVqxQenq6vvjiC8XHx1sdDwAQJijRAQBACErU0qXt9MwziVq1KkrPPVesAQNq9N57hera1Wd1OCDsONatU8rFFweNl996q8rvvtuCRACaK7/fr3Xr1um7777TVVddpUWLFiknJ0f33HOP0tPTFcXLxgAAFqBEBwAAISErK0LLl0epVy9JGqjly9N06aU1euihUrVrV3cwKAU60MRMU60vu0xRn3wSdCln/XoF0tIsCAWgOTJNU/fee68WLFig+Ph4DR06VJL0l7/8xeJkAABQogMAgBbuzTej9eqr0fr+e7vOPrtGXbvaJb2tiROvVc+ePa2OB4Qtx8aNShk8OGi8YtQolT30kAWJADQnJSUlWrZsmdxut7p3767x48frlFNO0bXXXqujjz5ahmFYHREAgD0o0QEAQIvh90sbNtQdDFpSYujJJ0vVqlVA48eX66yzauR0Sl9/zWpzwFKmqaSRI+VcujToUu7atfJ36mRBKADNwa5duxQXF6fCwkKdd955OvnkkzV48GANHDhQkjRs2DCLEwIAsH+U6AAAoFnzeKSqKkOtWpk666w2kqSMDI+GD6+SJA0e7LEyHoC92LduVZvzzgsarxw+XKVPPmlBIgBW+/nnnzVnzhxlZmZqx44dmjZtms466yx9/vnnatWqldXxAAA4IJToAACg2amsNLRkiVOLFjm1cmWUbrqpQrfeWqFZswrUrl1AvMIbaGZMU61uvlnR770XdCl31Sr5u3a1IBQAKwQCAW3YsEErV67U7bffri1btuinn37SuHHjdOaZZ8rlckkSBToAoEWhRAcAAM1CdrZNixc7NWiQRxUVht54I1oZGR7df3+ZOnTwS5Latw9YnBLAviK++05t+/ULGq++6CIVv/CCeNYLCB+TJ0/W9OnT5XQ6lZGRoerqag0cOHDPdi0AALRUlOgAAMBSy5ZF6amn4rR1q139+9coPb1WXbv6NGtWodXRAPyOhLvuUswbbwSN5y1dKl+PHhYkAtBUysrKtGLFCrndblVVVenVV19Vnz59dN5556lHjx4cDAoACCmU6AAAoMkEAtKXXzqUmenUmjVRmju3QCkpAd18c4X696+Ry2VaHRHAAYj4+We1Pe20oHHPueeq6LXXWH0OhKjc3Fz5fD6lpqaqX79+6tq1qzIyMjRo0CBJ0hlnnGFxQgAAGgclOgAAaFS1tVJuboQ6dvTr2muTtGWLXRkZHt17b5lsNqlXL6969fJaHRPAAYqfMEGxU6cGjecvXCjvCSdYkAhAYyosLNRbb70lt9utLVu26L777tPIkSO1du3aPfubAwAQ6ijRAQBAg/N6JbfbKbfbqeXLnTrvPI+mTCnRU0+VKDGRg0GBlsiWk6PUPn2CxmtOO02F777L6nMgRAQCAW3cuFFut1t/+tOf5Pf7tW3bNt14443q37+/oqOjJYkCHQAQVijRAQBAg8jPrzsYtFs3r3r39uqdd6J15pk1uvPOcnXuXHcwaFISB4MCLVHc448r7umng8YL3ntPtX37WpAIQEMyTVOGYWju3Ln6xz/+IcMwlJGRIUk64ogj9Mwzz1icEAAAa1GiAwCAw7J5s11/+1uCvv7aoTPPrNWRR/pkt0uvv15kdTQAh8lWWKjUXr2Cxr3HHaf8RYukiAgLUgH4f/buPC7Ken//+DXsuKAiIjJoLqjHxJRyX1JxwzItNTPTUrP1ZGaLx9KszNz7YR2t3MvyHMp9SwY1kxTNXBKX0tQsGXBFFAVkmfn90YlvOExuwM3yej4e/XG/75uZ65EjMhef+dz54cqVK9q8ebMsFot27typrVu3qlGjRlq4cKEaNWrEjUEBAPgLSnQAAHDD7HZp/353RUV5yWLx0iefXJC/v03Dhl1Rhw5XVa4cNwYFSopys2bJZ+JEh/m5yEhltGtnQCIAt+v8+fNKTExUSEiIhgwZooyMDIWHh+vll1+Wu7u7goODjY4IAECRRIkOAAD+VlaW9MsvbmrQIEtTppTXkiVlFB6errfeuqiaNbPk7i716JFudEwA+cSUnKxqDRs6zLPuuENnYmIkN95CAMVJWlqaFi1aJIvFon379mnQoEEKCQnR4sWL5e7ubnQ8AACKBX4CBgAAeYqO9tS6dd7auNFLNWpkad26cxo+/LJGjUqRi4vR6QAUhDILF6ri2LEO8/OffqqrXboYkAjAzbLb7Tp48KCioqLUvHlztWnTRkePHtWQIUPUsWNHlStXTpIo0AEAuAmU6AAAQJKUlOSiDRs85eYm9emTpthYT9Wvn6Xhw88qOPiPG4OWLct2LUBJZLp8WdXq13eYZ1eurNO7dkkeHgakAnCjsrKy5Orqqt27d+uf//ynMjMz1a1bN/n6+srV1VXTpk0zOiIAAMUaJToAAKWY3S5duGDS00/7au9ed7VqlaH+/VMlSW+/fcngdAAKg3dkpCq98orDPOmTT5T+wAMGJAJwI9LS0rRlyxZFRUVpw4YNWrlyperUqaOPP/5YTZo0kQsfGwMAIN9QogMAUIrY7dJPP7kpKspLUVHeeuqpy+rbN02DB1/RggVX5ePDSnOgtDClpSmgXj2ZbLZcc7uXlxIPHJC8vQ1KBsCZpKQkHThwQPfee6+mT5+u77//XuHh4XrhhRdybgpaqVIlg1MCAFDyUKIDAFDCZWdLu3d7qGnTDK1b56U336ygrl3TNXr0JbVpc1UmEzcGBUobr1Wr5Pv88w7zCxERSuvXz4BEAP7OokWLtGbNGu3Zs0f33nuv2rVrpzFjxrDaHACAQkKJDgBACbV9u4eWLvVWdLSXfHzsWrbsnMLD03X//encGBQora5eVUBoqFwuXnQ4lXjkiOxlyxoQCsBf2e12/fzzz4qKilLFihU1ZMgQJScna8CAAZo3b54qVKggSTKZTAYnBQCg9KBEBwCghEhONmnTJi/99purXn75so4dc1ONGtlauvS86tXLEu+1gdLNy2KR79ChDvPkSZOU+vjjBiQC8Kfs7GxlZ2crOTlZDz74oK5cuaKuXbuqZcuWkqQXX3zR4IQAAJRulOgAABRj2dmSi4s0dGglxcR4qWnTDPXokSZJGjgw1eB0AIqEzEz5t24tt4QEh1OJhw7J/r9VrQAKV3p6urZu3aqoqChFR0dr/Pjx6tWrlz788EOFhobK1dXV6IgAAOB/+DA3AADFiN0uHTnipg8/LKf77vPTO+/4yGSSnnrqinbvPqUvvzyvQYMozwH8wXPLFgXWrOlQoF98800lWK0U6EAhu3jxolatWiWbzSaLxaLp06crKChIX331lXr16iWTyaSmTZtSoAMAUMSwEh0AgCLOZpP27HFXrVrZSkkxqU+fyuraNV0vvZSidu2uSpJat84wOCWAIiU7W1U6d5b7kSMOp07t3y+br68BoYDSa+3atVq8eLF27type+65R23atFHPnj3Vq1cvo6MBAIAbQIkOAEARdfCgmz77rKw2bPCSp6ddH36YrObNM/Tjj6fFAjUAznjs2CG/Pn0c5pdefVWXR440IBFQ+hw9elTr169XYmKiJk6cKJvNpj59+uijjz5SpUqVjI4HAABuEiU6AABFREqKSd9846lvv/XS1KnJunTJRQEB2Vq8+LwaNPi/G4NSoAPIk80mv1695LFnj8OpU3v2yFa1qgGhgNLBZrPpypUrKleunO6//379/vvv6tKli+677z5JUs+ePQ1OCAAAbgclOgAABrp6VfL0lMaP99Fnn5VRaGimunVLV1aWSa1aZahVK7ZpAXB97nv3qkqPHg7zy889p0tjxxqQCCj5MjIyFBsbm3Nj0D59+mjMmDGaMWOGateuLTc33m4DAFBS8K86AACF7NgxV1ks3lq/3kseHnYtW3ZeDz+cqhdeuCxfX5vR8QAUJ3a7fAcMkFdMjMOp0zt3KttsNiAUUHKlpKRo8+bNat68udLT0zVhwgR1795dixYtUsOGDSVJ9erVMzglAADIb5To9j13ZgAAIABJREFUAAAUMJtNiotzl5eXXcHBWXr88cq6554MPf/8ZbVv/8eNQRs0yDI4JYDixu3gQfl37eowv/L447o4aZIBiYCSa8eOHZo1a5ZiY2PVuHFj1apVS40aNdLGjRuNjgYAAAoBJToAAAXEanXVrFnlZLF4yWSSRo++pH/8I0tbt57J2d8cAG6a3a5KTz8t76+/djh1ets2ZdesWfiZgBLm+PHjslgsiomJ0WeffSZPT0/df//9ioiIkJ+fn9HxAABAIaNEBwAgn1y5YtLmzZ6yWLw0YkSKypWzq2JFmxYuTFKjRpk5xTkFOoBb5Xb0qPzbt3eYp/bpo+QPPuAbDHCL7Ha7zp07p8qVK2vEiBHasGGDOnXqpIEDB0qSQkNDFRoaanBKAABgFEp0AADywbRpl/TJJ/XUqFGmwsPTVbGiXX5+No0alWJ0NAAlgd2uiiNHqsySJQ6nzmzerCz2YAZuWnZ2tmJjY2WxWLRhwwbVrVtXX3zxhUaNGqXp06fL3d3d6IgAAKCIoEQHACAfWK2zFRr6m7788gO5uroaHQdACeL622+q2rq1wzyte3ddmDuX1efATUhNTdW3336rgIAANW7cWBEREWrRooWWL18u8/9uxGvmhrwAAOAaLkYHAACgJJgy5UWlp5/Q1KlTjY4CoASpMGZMngX6GYtFF+bNo0AHbtCxY8c0ZMgQ3XXXXZo9e7aSkpLk6uqq5cuXa/To0WrWrJlM/H0CAABOUKIDAJAPPD09NXfuXJ08eVKZmZlGxwFQzLlYrQo0m1X2009zza+2bauE+HhlhYQYEwwoJn7//XfNnTtXffv21c8//6wKFSqoU6dOio2N1apVq9S5c2ejIwIAgGKEEh0AgHxSrVo1ffTRR0pNTdXx48eNjgOgmCo/caICmjd3mJ9ds0bnv/yS1edAHux2u06ePClJ+uijj9S1a1ft3btXgwYNUvXq1eXn56eBAwfK39/f4KQAAKA4Yk90AADymcVi0QcffKB169apYsWKRscBUEy4nDmjgNBQh3lGaKjOrV4tubD+BbjWDz/8oLVr18pisUiStm7dqgEDBujJJ5+Up6enwekAAEBJwU/iAADks4cfflgtWrTQ8OHDlZ2dbXQcAMVAuYiIPAv0c0uX6tzatRTowP+kpaUpOjpaa9askSQtW7ZMHh4emjlzpmJjY+Xm5qaKFStSoAMAgHzFSnQAAPKZyWTSxIkT1bt3b61fv149evQwOhKAIsqUlKRqjRo5zDPr1tXZTZskV1cDUgFFz+XLlzVy5Eh9++23qlevnh577DFJ0uTJkw1OBgAASgNKdAAACoCXl5e++uorlS1bVpmZmXJ3dzc6EoAipuzs2aowfrzD/PzixbraoUPhBwKKEKvVqujoaEVFRalv377q27evOnXqpHfeeUeBgYFGxwMAAKUMnwsFAKCAlCtXTqmpqerYsaOOHDlidBwARYTp0iUFms0OBXp2tWpKOHGCAh2lkt1u1+HDh2Wz2bRp0yZ16NBBsbGx6tevn7p06SKTyaT+/ftToAMAAEOwEh0AgAJUtmxZDR06VEOHDtXXX38tHx8foyMBMFCZzz9XxdGjHeZJ8+crPTzcgESAsQ4cOKAVK1YoKipKKSkpWrNmjdq2bau4uDh5e3sbHQ8AAEASJToAAAVuyJAh2rdvn0aMGKEFCxbIZDIZHQlAITOlpqpa3boOc1v58jr144+Sl5cBqYDCd/XqVW3btk2HDx/Wc889p127dslms+n//b//p6ZNm8qV+wAAAIAiiBIdAIACZjKZNHnyZO3atYsCHSiFvJcuVaURIxzmF2bNUtqDDxqQCChcdrtdJpNJr776qtasWaOaNWvq/vvvl91u1+DBg42OBwAAcF2U6AAAFAJvb2+1a9dOK1euVJkyZdS1a1ejIwEoaOnpCmjYUC7p6bnGdpNJp375RXa2qkAJdvr0aVksFlksFtWuXVvvvvuuwsPDNWLECFWvXt3oeAAAADeFG4sCAFCIqlSpopEjR+ro0aNGRwFQgLzWrVNgnToOBXry9OlKjI+nQEeJdOzYMSUlJSkhIUFt27bVli1b1KtXL40cOVKS1LlzZwp0AABQLLESHQCAQtSmTRu99NJLGjp0qNatW6fy5csbHQlAfsrIUNVmzeR67pzDqcSff5adv/MoYX799VdFRkbKYrHo7NmzmjVrljp06MCNQQEAQInCSnQAAArZsGHD1KVLF506dcroKADykefGjQqsVcuhQL84frwSrFYKdJQIGRkZiomJ0bhx45SRkaHExESlpqbqvffe0759+9ShQwdJokAHAAAlCivRAQAoZCaTSW+++aZsNpu2b9+uVq1aGR0JwO3IypJ/+/ZyO3HC4VTigQOyV6pU+JmAfGSz2eTi4qIZM2Zo9uzZMpvN6tatm65evarWrVurdevWRkcEAAAoUJToAAAY5MyZM3rmmWf0/vvvq0uXLkbHAXALPLZuld8jjzjML73+ui6/8IIBiYD8cfbsWW3YsEFRUVG6dOmSVq5cqQ4dOqhXr16qVauW0fEAAAAKFSU6AAAGCQgI0MyZM/Xcc89p9erVqlOnjtGRANwom01VwsPlfvCgw6lT+/bJ5udnQCjg9pw4cUJZWVmqXbu2evToofr16ys8PDznF71NmjQxOCEAAIAxKNEBADDQvffeq+HDhysiIkIzZ840Og6AG+D+ww+q8uCDDvOUF19Uyr/+ZUAi4NadO3dOCxculMVikdVq1b/+9S8FBwdr27ZtcnPj7SIAAIBEiQ4AgOGeeeYZZWRkKD09XZ6enjKZTEZHApAXu12V+/aV544dDqdO7dolW7VqBoQCbk5WVpa+//57WSwWDRo0SBUqVFBSUpLefPNNtWrVSh4eHpJEgQ4AAPAXLkYHAACgtDOZTPL09NSoUaNYjQ4UUe5xcQoMCnIo0C8/+aQSrFYKdBRpmZmZkqQVK1aocePGeuONN+Tt7a0yZcrI399fkyZNUvv27XMKdAAAAOTG8gIAAIqIV155RT169FBISIg6duxodBwAkmS3y3fwYHlt3Ohw6vT27cquUcOAUMD1JSUlaePGjbJYLNq5c6d27typpk2batWqVQoODjY6HgAAQLHCSnQAAIqIO+64QzNnztTw4cOVkJBgdByg1HP7+WcFBgU5FOhXHn1UCVYrBTqKnPj4eO3cuVOS9Nprr2nVqlVq3769NmzYIG9vb1WvXp0CHQAA4BawEh0AgCKkffv2WrBggQICAoyOApRedrsq/vOfKrNqlcOp0zExyq5Tx4BQQN6uXr2qjz76SFFRUTpx4oQee+wxNW/eXPPmzeMeGwAAAPmEEh0AgCKmefPmOnr0qBYtWqR33nmHEgQoRK7Hjqnqvfc6zNN69tSFjz6S+PsIg2VnZ2vXrl2KiopSs2bN1L17d125ckWjRo1S27Zt5enpKUn82wEAAJCP2M4FAIAiqFq1atq6das+/vhjo6MApUaFUaPyLNDPbNyoCx9/TIEOw6Snp8tms2nv3r0KDQ3Vyy+/LBcXF9WuXVsmk0ljx45Vp06dcgp0AAAA5C9WogMAUASVLVtW8+fP1wMPPKCGDRuqffv2RkcCSizXkydVtWVLh3l6WJiSFi2iPIchkpOT9c0332j9+vXasmWLli9frnr16mnJkiWqV68eK80BAAAKESvRAQAoomrVqqUPP/xQSUlJRkcBSiyft9/Os0A/u369kj7/nAIdhSohIUFff/21JGnhwoWKjIxUq1attHnzZoWEhKhs2bKqX78+BToAAEAhYyU6AABFWFhYmCQpJiZGTZs2VZkyZQxOBJQMLomJCmja1GF+tWVLnV+6lPIchWrevHlasWKFjhw5oo4dOyo8PFwvvfSSRo4caXQ0AAAAiBIdAIBi4fPPP1dkZKRmzZrFCkTgNpWfNk3lZ8xwmJ9buVIZzZoZkAilyZ97m0dFRals2bJ66aWX5ObmphEjRqhdu3by9vY2OiIAAACuQYkOAEAxEBERoR49emj27Nl69tlnjY4DFEsu58+raqNGDvPMhg11dv16ydXVgFQoDTIyMpSVlaWMjAx17NhRHh4eCg8PV3h4uCRp8ODBxgYEAADA32JPdAAAioFy5cpp/vz5Wrx4sVJSUoyOAxQ/U6bkWaCfi4zU2ehoCnTku8uXL2vVqlV6/vnnddddd2nt2rWqWLGiIiMjtWPHDr3zzju65557jI4JAACAG8BKdAAAiok6depo8+bNcnV1VUpKisqXL290JKDIMyUnq1rDhg7zrDvu0JmYGMmNH4eRf86cOaONGzeqb9++2rlzpxYtWqTw8HCNHj1aNWrUkCTVr1/f4JQAAAC4WaxEBwCgGHFzc5PFYlGfPn2UlpZmdBygSCuzcGGeBfr5Tz/VmdhYCnTkm1WrVqlXr15q3bq1oqOjdeHCBYWFhWnZsmV66qmncgp0AAAAFE+8cwAAoJjp2rWrvvzyS40aNUoffvghNxoFrmG6fFnV8ljtm+3np9M//CB5eBiQCiWF3W5XXFyc1q9fr99++00ff/yxypUrp2eeeUbt27dX2bJljY4IAACAfMZKdAAAihkXFxd98MEH2rdvn/773/8aHQcoUrwjI/Ms0PXVVzoTF0eBjluSmZmp8+fPS5K6d++uJ598UpcvX9aAAQMkSZ06ddJ9991HgQ4AAFBCsRIdQIm3YsUK7dy5U1arVR4eHqpXr54GDhyowMBAo6MBt8zHx0eff/65KlasaHQUoEgwpaUpoF49mWy2XHO7l5dOHTyoarVrS4mJBqVDcZSWlqZvvvlGUVFR2rRpk/r3769x48Zp/vz5CgwM5FNAAAAApQgr0QGUeIcOHVK3bt303nvvaezYsbLZbJowYYLS09ONjgbcljvuuENly5ZVv379FBcXZ3QcwDDeK1eqWnCwQ4F+YcYMJR47Jnl7G5QMxc358+cVGRmpEydOKCUlRXPmzNGdd96pNWvWaNy4cZIks9lMgQ4AAFDKsBIdQIk3ZsyYXMfPP/+8hg0bpuPHj+vOO+80KBWQP9zc3BQYGKgePXpo8uTJevTRRyl3UHpcvaqAJk3kcumSw6nEI0dkZ2sN3KDt27dr2rRp2rdvn1q3bq2GDRuqZs2aWrVqldHRAAAAUARQogModVJTUyVJ5cqVy/N8ZmamMjMzc45NJpO8/7eKkXKy6Pnzz6Q0/9lMmTJFy5Yt02uvvab169dr8uTJCgoKMjqWYf76mijNr4uSzjMqSr5DhzrML06erNTHH5ck/fmnz/cJXOvHH3/UF198oU2bNmnJkiWqXLmyBg8erLCwMJUvX97oeChkfI/AtXhNAACuZbLb7XajQwBAYbHb7Zo6daquXLmi8ePH53nNV199paVLl+Yc16pVS1OmTCmsiMAt6dy5szZt2iTpj18QTZ06Vc8884xcXErfzm179uzRPffco927d+vuu+82Og7yW2amVLu2FB/veO7CBYn7BCAPWVlZio+PV82aNfXEE08oOjpavXr10kMPPaSwsDC5u7sbHREAAABFGCU6gFJl3rx52rt3r8aPH6/KlSvneY2zlehnz55VVlZWYUXFDTKZTAoICNCpU6dUmv9JmzVrlt57771cs5YtW2r69OmqXbu2QamMsX//fnXr1k0Wi0WNGjUyOg7ykce336rygAEO80vjxunKs886/Tq+T5RONptNGzZsUFRUlKKjoxUaGqovvvhCSUlJatCggc6cOcPrAZL4HgFHpe014ebmpipVqhgdAwCKNLZzAVBqLFiwQLt379Y777zjtECXJHd3d6cr0krDD9HFld1uL9V/Pm3atHGY7dixQ507d9arr76qp556Sm5upeOf/T9fB6X9NVGiZGerSufOcj9yxOHUqf37ZfP1lW7gz5rXRMl34cIFbdq0SQEBAWrTpo0iIyMVGhqqlStXKjg4WHa7Xb6+vnJxceH1AAe8JnAtXhMAgD+Vvs94Ayh17Ha75s+fr++//17jxo2Tv7+/0ZGAfBcSEqIKFSo4zNPT0zVhwgT17NlTP/30kwHJgNvjsWOHAmvUcCjQL736qhKs1j8KdJR6x48fV79+/dS0aVMtW7ZMGRkZMplMWrBggYYPH67g4GCjIwIAAKAYo0QHUOLNnz9f3333nUaMGCFvb28lJycrOTlZGRkZRkcD8o2rq6tatWrl9Py+ffsUHh6u6dOn89pH8WCzye+BB+TXp4/DqVN79ujyyJEGhEJRYLfb9fPPP2vGjBkKDw/Xrl275Ofnp/79+2vPnj3673//q7CwMKNjAgAAoAShRAdQ4kVHRys1NVVvv/22nn766Zz/YmNjjY4G5Ku8tnT5q6ysLEVERCg8PFx79+4tpFTAzXPfu1eB1avLY8+eXPOU55//Y/V51aoGJYNRsrOzdejQIUnS7Nmz1a9fP8XHx+uVV15RSEiIfHx81Lt37zw/kQMAAADcrtKxOSqAUu2rr74yOgJQKK5Xov/p8OHD6tmzp4YNG6ZRo0bJ29u7gJMBN8hul++AAfKKiXE4dWrnTtnMZgNCwUgxMTFau3atLBaLypcvr02bNunxxx/XU089JVdXV6PjAQAAoJRgJToAACVEvXr15Ofnd0PX2mw2zZkzR507d+ZTGSgS3A4eVGBQkEOBfuWJJ/5YfU6BXipcunRJK1eu1H//+19JUmxsrKpVq6bIyEh999138vT0VJkyZSjQAQAAUKgo0QEAKCFMJtMNr0b/04kTJ/Twww9r1KhRunTpUgElA/6G3a5Kw4bJv2tXh1Ont23TxYkTDQiFwmS325WWlqYBAwYoNDRUX3zxhVxc/nibMnr0aI0cOVINGjSQyWQyOCkAAABKK0p0AABKkLZt297S1y1evFgdO3bUxo0b8zkR4JzbL78oMChI3uvX55qn9u2rhPh4ZdesaUwwFLijR49q5syZ6tGjhxYuXChvb2899thj+uGHH7R06VI98sgjRkcEAAAAclCiAwBQgtzsSvS/OnXqlJ544gkNHz5cSUlJ+ZgKuIbdroovvij/Dh0cTp359lslf/CBxKrjEsVms2nv3r3KyMhQTEyMevXqpaNHj+qFF17Qo48+Kkm6//775evra3BSAAAAwBE3FgUAoASpUaOGgoKCFB8ff8uPsXz5cm3ZskXvvvuuevbsyRYKyFeuJ06oah6/7Enr3l0X5s6lPC9h9u7dq6+++krR0dFydXXVf/7zH7Vu3Vr79u2TmxtvRQAAAFA8sBIdAIAS5Fb2Rc/L+fPn9fzzz+vJJ5/UqVOn8iEZIFUYMybPAv2MxaIL8+ZRoJcAly9f1po1azRp0iRJ0vHjx1W5cmV99tln+v777xUcHCw3NzcKdAAAABQrlOgAAJQw+VGi/2n79u3697//rYyMjHx7TJQ+LlarAs1mlf3001zzq23bKiE+XlkhIcYEQ76w2WySpJEjR6px48ZasGCBfH19Zbfb1adPH7366qsKCQnhUy0AAAAotlgCAgBACXO7JXpAQIC6deum8PBwtWzZUh4eHvmUDKVR+YkTVX7WLIf52TVrlHn33QYkQn44ceKEoqKiZLFYdMcdd2jGjBnq37+/xowZIz8/P6PjAQAAAPmKEh0AgBImICBAwcHBOnr06A1/TXBwsMLDwxUeHq7GjRvLxYUPq+H2uJw5o4DQUId5Rmiozq1eLfEaK1bsdrv279+vKlWqyMPDQ/fff7/CwsL05JNPqsP/bhDbokULY0MCAAAABYQSHQCAEqhNmzY3VaK3aNFCr7/+egEmQmlSLiJCPtOnO8zPLV2qjFatDEiEW3X06FF99tlnioqKks1m07Rp0xQWFqa4uDi5uroaHQ8AAAAoFCwBAgCgBMprSxd3d3d16NBBkydP1p49e1SxYsWcc4sXL1Z0dHRhRkQJZEpKUqDZ7FCgZ9arp4Tff6dALwZSU1O1fv16jRw5UleuXNHly5dVpkwZzZkzRz/88IPCwsIkiQIdAAAApQor0QEAKIFatWolk8mksmXLKiwsTOHh4erYsaN8fHxyrtm/f7+qV6+eczxkyBDt2rVL1apVMyIyirmys2erwvjxDvPzixfr6v+2+0DRlJGRIQ8PD0VERGjWrFlq0KCBwsPDlZ2drSZNmqhJkyZGRwQAAAAMRYkOAEAJ5Ovrq2XLlqlJkyby9PTM8xoXFxft3btXoX/Zt7pp06b6/fffWWWKG2a6dEnVGjRwmGdXq6bT27dL7u4GpML1nDx5UhaLRVFRUUpOTtbGjRt1//3369FHH1VAQIDR8QAAAIAihe1cAAAooVq0aOG0QP+Tv7+/vvjii1yz+vXrF2QslCBlPv88zwI9af58nd61iwK9CLHb7Tp06JAOHDggu92e88mTQYMGafny5ZKkevXqUaADAAAAeWAlOgAApVzHjh01dOhQLViwQJKUlpamsWPHasKECQYnQ1FlunJF1erVc5jbfHx06scfpev88gaF59y5c5o1a5YsFotSU1M1cuRIhYSEaMOGDTKZTEbHAwAAAIoFVqIDAAC9++678vb2zjleuHChNm/ebGAiFFXeS5bkWaBfmDVLp376iQLdYGlpadqwYYNeffVVxcXFycPDQy4uLvrggw+0e/duPfHEE5JEgQ4AAADcBFaiAwAASdLhw4dVo0aNnOOBAwdq79698vf3NzAVioy0NFULCZEpPT3X2O7iolNHjsj+l1/CoHClpqaqTJkyWrVqlV577TUFBwerW7du8vPzk4+Pj958802jIwIAAADFGiU6AACQJLm6umr37t265557cmahoaE6efKkXFz48Fpp5rV2rXyfecZhnjx9ulIffdSAREhISFB0dLSioqK0Z88e7d69W61atdLmzZtlNpuNjgcAAACUKLwjBgAAOQICAvTpp5/mmoWEhBgTBsbLyFDVxo3zLNATf/6ZAr0Q2e12HTlyRN98840kadq0adq2bZv69u2r77//XuXLl5e/vz8FOgAAAFAAKNEBAEAuXbp00cCBA3OOL168qLffftu4QDCE58aNCqxVS67nzuWaXxw/XglWq+zlyxuUrHTJysrSe++9p3bt2ql3797avn27JCkiIkJz585V3759ValSJYNTAgAAACUb27kAAAAHU6ZMUWRkpLKysiRJc+fOVVhYmO69916Dk6HAZWXJv317uZ044XAq8cAB2SlsC9TVq1cVGxur9evX65577tEjjzwiHx8fTZ8+XU2bNpWbGz++AwAAAIWNlegAACBPx44dy3X86KOP6tw1q5JRsnhs3arAO+5wKNAvvf76H6vPKdALREpKijIzM7V//37dddddmjhxogICAnLuTzB8+HC1bNmSAh0AAAAwCD+JAwCAPLm5uWnnzp1q3rx5zqxx48bcaLQkstlUJTxc7gcPOpw6tW+fbH5+BoQq2c6ePauoqChZLBZt375dS5YsUcOGDbVhwwbVqFHD6HgAAAAA/oJ3wAAAwCmz2az58+fnmv25OhYlg/sPPyiwenWHAj1lxAglWK0U6Pno2LFjioyMlCStXbtW33zzjR544AHt3LlTd999tzw9PSnQAQAAgCKIEh0AAPyt8PBw9evXL+f4zJkzmjRpkoGJkC/sdlXu00dVHnzQ4dSpXbuUMmqUAaFKpk8++UQdO3ZUz549tX37dmVmZmrIkCFauHChHnnkEVWuXNnoiAAAAAD+BiU6AAC4roiIiFzHM2fO1LZt2wxKg9vlHhenwKAgee7YkWt+ediwP1afV6tmULLiLzMzUzExMRozZozeeecdSZK/v7/Gjx+vH3/8UR988IHc3d0NTgkAAADgZrAnOgAAuCEnTpxQzZo1c4779eun/fv3y9fX17hQuDl2u3yfeEJemzY5nDq9Y4eyq1c3IFTxl5qaqszMTLm5ual58+YKCAhQt27ddN9990mSevfubXBCAAAAALeDEh0AANwQd3d3ff/992rRokXOrFGjRtxotJhw+/ln+Xfq5DC/MmCALk6bZkCi4i05OVlRUVGKiorS1q1bNWHCBPXv31/R0dEym81GxwMAAACQj3jHCwAAblhQUJBmz56da9ayZUuD0uCG2O2q9NxzeRbop2NiKNBvwm+//abZs2frypUr+uWXX7Ru3Tp16dJFsbGx6t+/vyRRoAMAAAAlECU6AAC4KT169Mi1PYXVatU0itgiyfXYMQUGBcl79epc87SePZUQH6/sOnUMSla8rFixQp07d1a3bt0UFxenixcvqlmzZvr888/12GOPyd/f3+iIAAAAAAoQ27kAAICb9u9//1vLly/POZ4xY4buvffeXFu9wFgVRo1S2cWLHeZnNm5UVoMGBiQqHrKysrRz505FRUXp8OHDioyMVGBgoMaOHavWrVvLw8PD6IgAAAAAChklOgAAuCXX3mi0d+/eOnDggCpVqmRcKMj15ElVzWOLnfSwMCUtWiSZTAakKtrS0tKUlJQks9msBx54QJcvX1b37t01atQoSeKXQwAAAEApR4kOAABuibu7u2JjY9W6deucWUhIiOLj42WiqDWEz9tvq9zcuQ7zs+vXK/OuuwxIVHSlpaVp7dq1ioqK0pYtW/T4449r3LhxWrx4sXx9fY2OBwAAAKAIYU90AABwy+644w7NmjUr16xt27YGpSm9XBITFWg2OxToV1u2VEJ8fKkr0O12uy5evKgjR44oJiZGcXFxkqT4+HgtWLBAP/30kzIzM7Vy5Uq1a9dO3333ncaNGydJFOgAAAAAHLASHQAA3JYHH3xQX3/9tdatWyfpj21eIiIiNHLkSIOTlQ7lp01T+RkzHObnVq5URrNmBiQqWFeuXNGpU6d0+vTpnP/yOk5PT8/5mlq1aqlSpUo6fPiwOnbsqGbNmsnHx0eL89gzHgAAAACuRYkOAABu25w5c2Q2m3OOp0+frrZt26pZCSxxiwqX8+cVkMcK84yQEJ1bv15yKV4fOExPT9eZM2d06tTlGY4pAAAcHklEQVSpvy3JL1++fNOP/euvv+r1119X586d5enpWQDpAQAAAJRklOgAACBf/Prrr6pVq1bO8YMPPqhDhw6pQoUKBqYqmcrNmiWfiRMd5uciI5XRrp0BiW6M1WrVhg0b8izJk5OTC+x527Vrp/DwcLm5uclutxfY8wAAAAAomSjRAQBAvvDw8NB3332ndn8pce+8805uNJqPTMnJqtawocM8q2ZNndmyRXIr2j/a+fn5afny5dq9e3ehPF9AQIDeeust9ezZU4GBgUpMTCyU5wUAAABQshSvz/kCAIAirXbt2vrggw9yzTp16mRQmpKlzMKFeRbo5z/7TGe2bSvyBbokeXp6as6cOfL39y/Q53Fzc9Ozzz6rLVu2qGfPnvwSBwAAAMBtoUQHAAD5qm/fvurWrVvO8eHDhzVz5kwDExVvppQUBZrNqjh2bK55tp+fEn79VVc7dzYo2a0JCAjQnDlz5O7uXiCP36pVK0VHR+vNN99UuXLlCuQ5AAAAAJQulOgAACDfLViwINfxpEmTtGfPHoPSFF/ekZGq9o9/OMyTZs/W6X37JA8PA1LdvmbNmmn8+PH5+pj+/v6aOXOmlixZovr16+frYwMAAAAo3SjRAQBAgTh27Fiu4wceeEApKSkGpSleTGlpqla9uiq98kquud3LSwlHjyq9Rw+DkuWfRx55RB758EsAV1dXDRs2TFu2bNFDDz3E1i0AAAAA8h0lOgAAKBBeXl7asmVLrtk//vEP2e12gxIVD94rV6pacLBMNluu+YUZM5R47Jjk7W1Qstt34cIFvfrqqzKbzapdu7YyMjJu6/GaNWum9evX65133pGPj08+pQQAAACA3Ir+HagAAECxFRwcrPfff1+v/GVFdXh4uCwWi4GpiqirVxXQpIlcLl1yOJV45IjsZcsaEOr2nThxQq+//rpiYmLy7TErV66ssWPHqm/fvnJxYU0IAAAAgILFuw4AAFCg+vfvr44dO+YcHzhwQJ988omBiYoeL4tFgbVrOxToyZMmKcFqLXYF+g8//KB27drJbDarTZs2f1ugh4WFaebMmXJzu/7aDhcXFw0ePFgxMTHq168fBToAAACAQsFKdAAAUOC++OILmc3mnON3331XrVu31l133WVgqiIgM1NVW7WSa2Kiw6nEQ4dkr1DBgFA3z263a/Xq1Xr55ZeVnp5+3esHDRqk0aNHq2LFijmzixcvasyYMU6/JjQ0VJMmTVKjRo3yJTMAAAAA3CiW7wAAgEJx9OjRXMfdu3fX5cuXDUpjPM8tWxRYs6ZDgX5x3Lg/Vp8X8QI9MzNTH3/8scxms4KCgvT888//bYH++uuv69dff5XVatXkyZNzFeiS9MQTT6hfv34OX1epUiVNmzZNq1evpkAHAAAAYAhWogMAgELh7e2tb775RmFhYTmz+vXrKz4+XiaTycBkhSw7W1U6d5b7kSMOpxL375fd19eAUDfm0qVLmjp1qhYuXHjda93c3BQREaGHHnrohv58TSaTJk2apMOHD2vfvn0ymUwaMGCARo8eLd8i/P8EAAAAQMnHSnQAAFBo6tevr8mTJ+ea9ezZ06A0hc9j+3YF1qjhUKBfevXVP1afF8Gy2Gq1avDgwTKbzWrQoMHfFug1a9bUsmXLZLVa9dtvv6l379439QsSLy8vzZ07Vx06dNCaNWs0depUCnQAAAAAhmMlOgAAKFSDBg3S6tWrFRsbK0nas2eP5s2bp2HDhhmcrADZbPLr2VMee/c6nDq1d69s/v4GhHIuLi5Or7zyig4dOnTda1u3bq1JkyYpODg4X57bbDZr8eLF+fJYAAAAAJAfWIkOAAAK3VdffZXr+K233tKBAwcMSlOw3PfsUWD16g4FesrzzyvBai0SBbrdbpfFYtGdd94ps9ms7t27/22B/vDDDysuLk5Wq1VLlizJtwIdAAAAAIoiVqIDAIBCZzKZdPTo0Vzla7du3fTLL7+oTJkyBibLR3a7fAcMkFd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      "text/plain": [
       "<Figure size 1500x500 with 3 Axes>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "# This code cell will not be shown in the HTML version of this notebook\n",
    "vec1 = np.asarray([2,1])\n",
    "vec2 = np.asarray([-1,2])\n",
    "alpha1 = 4\n",
    "alpha2 = 1.5\n",
    "\n",
    "plotter = vector_plots.vector_linear_combination_plot(vec1, vec2, alpha1, alpha2)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "id": "qKrLPRoBe4MN"
   },
   "source": [
    "Indeed by changing the values of $\\alpha_{1}$ and $\\alpha_{2}$ in $\\alpha_{1}\\mathbf{x}_{1}+\\alpha_{2}\\mathbf{x}_{2}$ each time we get a new vector (or point) that is a linear combination of $\\mathbf{x}_{1}$ and $\\mathbf{x}_{2}$. "
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {
    "id": "nYyKVdk2e4MN"
   },
   "outputs": [],
   "source": [
    "# # animate a two-dimensional spanning set ranging over a coarse set of points in the space - uncomment to re-render animation\n",
    "# C = np.array([[2,-1],[1,2]])\n",
    "# linlib.span_animation.perfect_visualize(video_path_2,C=C,num_frames = 200,fps=10)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {
    "id": "Y3zanChDe4MN",
    "outputId": "fffcec35-0b7d-4ce5-f0f8-2349fa2cad96"
   },
   "outputs": [
    {
     "data": {
      "text/html": [
       "<video width=1000 controls><source 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      ],
      "text/plain": [
       "<IPython.core.display.HTML object>"
      ]
     },
     "execution_count": 3,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "show_video(video_path_2)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "id": "6QvtBL7Je4MN"
   },
   "source": [
    "The set of all such vectors/points - created by taking a linear combination of vectors $\\mathbf{x}_{1}$ and $\\mathbf{x}_{2}$ - is referred to as the span of $\\mathbf{x}_{1}$ and $\\mathbf{x}_{2}$. In the example above the span of vectors $\\mathbf{x}_{1}$ and $\\mathbf{x}_{2}$ is the entire 2D plane, or put in different words, $\\mathbf{x}_{1}$ and $\\mathbf{x}_{2}$ span the 2D plane.\n",
    "\n",
    "But isn't the span of two $1\\times 2$ vectors always the entire 2D plane? Not necessarily! Take the two vectors $\\mathbf{x}_{1} = [2, \\, \\, 1]$ and $\\mathbf{x}_{2} = [4, \\, \\, 2]$ for example. Because these two vectors point at the same direction (one is a scalar multiple of the other), any linear combination of the two will have the same direction. Let's verify this by looking at a couple of examples.    "
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {
    "id": "uBrKPiDGe4MN",
    "outputId": "3c0a1e31-da0b-4b99-c8dc-0c2ef2d043ec"
   },
   "outputs": [
    {
     "data": {
      "image/png": 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",
      "text/plain": [
       "<Figure size 1500x500 with 3 Axes>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "## This code cell will not be shown in the HTML version of this notebook\n",
    "vec1 = np.asarray([2,1])\n",
    "vec2 = np.asarray([4,2])\n",
    "alpha1 = .5\n",
    "alpha2 = 2\n",
    "\n",
    "plotter = vector_plots.vector_linear_combination_plot(vec1, vec2, alpha1, alpha2)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {
    "id": "LvLcK7V-e4MO",
    "outputId": "97168362-9ccf-42ff-ac2d-b8a7bf13f2e5"
   },
   "outputs": [
    {
     "data": {
      "image/png": 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      "text/plain": [
       "<Figure size 1500x500 with 3 Axes>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "## This code cell will not be shown in the HTML version of this notebook\n",
    "vec1 = np.asarray([2,1])\n",
    "vec2 = np.asarray([4,2])\n",
    "alpha1 = 1\n",
    "alpha2 = -2\n",
    "\n",
    "plotter = vector_plots.vector_linear_combination_plot(vec1, vec2, alpha1, alpha2)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "id": "c_Ym8gaEe4MO"
   },
   "source": [
    "So in this case the linear combination of $\\mathbf{x}_{1} = [2, \\, \\, 1]$ and $\\mathbf{x}_{2} = [4, \\, \\, 2]$ is just a line that can be traced out using scalar multiples of any of the two vectors. In other words, given either one of $\\mathbf{x}_{1}$ or $\\mathbf{x}_{2}$ the other one becomes redundant. In linear algebra jargon, such vectors are called linearly dependent. \n",
    "\n",
    "The notion of linear combination of vectors can be extended in general to a set of $k$ vectors $\\left\\{\\mathbf{x}_{1}, \\mathbf{x}_{2}, \\ldots,\\mathbf{x}_{k}\\right\\}$, all of the same dimension, taking the form\n",
    "\n",
    "$$\\sum_{i=1}^k \\alpha_{i}\\mathbf{x}_{i}=\\alpha_{1}\\mathbf{x}_{1}+\\alpha_{2}\\mathbf{x}_{2}+\\cdots+\\alpha_{k}\\mathbf{x}_{k}$$\n",
    "\n",
    "If these vectors span a k-dimensional space they are called linearly independent. Otherwise there is at least one vector in the set that can be written as a linear combination of the rest. "
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "id": "u2AvxAD6e4MO"
   },
   "source": [
    "#  Matrices and matrix operations"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "id": "f1VOCXs9e4MO"
   },
   "source": [
    "In this Section we introduce the concept of a matrix as well as the basic operations one can perform on a single matrix or pairs of matrices.  These completely mirror those of the vector, including the transpose operation, addition/subtraction, and several multiplication operations including the inner, outer, and element-wise products.  Because of the close similarity to vectors this Section is much more terse than the previous. "
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "id": "JlI7fdsxe4MO"
   },
   "source": [
    "##  Basic ideas"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "id": "xXppNCr0e4MO"
   },
   "source": [
    "If we take a set of $P$ row vectors - each of dimension $1\\times N$ "
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "id": "kFZB1N91e4MP"
   },
   "source": [
    "$$\\mathbf{x}_{1}=\\left[\\begin{array}{cccc}\n",
    "x_{11} & x_{12} & \\cdots & x_{1N}\\end{array}\\right]$$\n",
    "\n",
    "$$\\mathbf{x}_{2}=\\left[\\begin{array}{cccc}\n",
    "x_{21} & x_{22} & \\cdots & x_{2N}\\end{array}\\right]$$\n",
    "\n",
    "$$\\vdots$$\n",
    "\n",
    "$$\\mathbf{x}_{P}=\\left[\\begin{array}{cccc}\n",
    "x_{P1} & x_{P2} & \\cdots & x_{PN}\\end{array}\\right]$$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "id": "qPkWrusve4MP"
   },
   "source": [
    "and stack them one-by-one on top of each other we form a *matrix* of dimension $P\\times N$\n",
    "\n",
    "\n",
    "$$\n",
    "\\mathbf{X}= \\begin{bmatrix}\n",
    "x_{11} & x_{12} & \\cdots & x_{1N}\\\\\n",
    "x_{21} & x_{22} & \\cdots & x_{2N}\\\\\n",
    "\\vdots & \\vdots & \\ddots & \\vdots\\\\\n",
    "x_{P1} & x_{P2} & \\cdots & x_{PN}\n",
    "\\end{bmatrix}\n",
    "$$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "id": "bRFU1wYqe4MP"
   },
   "source": [
    "In interpreting the dimension $P\\times N$ the first number $P$ is the number of rows in the matrix, with the second number $N$ denoting the number of columns.\n",
    "\n",
    "The notation we use to describe a matrix is a bold uppercase letter, as with $\\mathbf{X}$ above.  Like the vector notation nothing about the dimensions of the matrix is detailed by its notation - we need to explicitly state these."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "id": "D9AW7oAre4MP"
   },
   "source": [
    "The *transpose* operation we originally saw for vectors is defined by extension for matrices.  When performed on a matrix the transpose operation flips the entire array around - every column is turned into a row, and then these rows are stacked one on top of the other forming a $N\\times P$ matrix.  The same notation used previously for vectors - a superscript $T$ - is used to denote the transpose of a matrix\n",
    "\n",
    "$$\n",
    "\\mathbf{X} ^T= \\begin{bmatrix}\n",
    "x_{11} & x_{21} & \\cdots & x_{P1}\\\\\n",
    "x_{12} & x_{22} & \\cdots & x_{P2}\\\\\n",
    "\\vdots & \\vdots & \\ddots & \\vdots\\\\\n",
    "x_{1N} & x_{2N} & \\cdots & x_{PN}\n",
    "\\end{bmatrix}\n",
    "$$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "id": "1DcrUxfRe4MP"
   },
   "source": [
    "In numpy we define matrices just as we do with arrays, and the same notation is used to transpose the matrix.  We illustrate this with an example in the next  Python cell."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {
    "id": "XCpxeZDQe4MP",
    "outputId": "9c41d273-71b2-4ac5-e9e8-981d2c8b4d25"
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "----- the matrix X -----\n",
      "[[1 3 1]\n",
      " [2 5 1]]\n",
      "----- the transpose matrix X^T -----\n",
      "[[1 2]\n",
      " [3 5]\n",
      " [1 1]]\n"
     ]
    }
   ],
   "source": [
    "# create a 2x3 matrix\n",
    "X = np.array([[1,3,1],[2,5,1]])\n",
    "print ('----- the matrix X -----')\n",
    "print (X) \n",
    "\n",
    "# transpose the matrix\n",
    "print ('----- the transpose matrix X^T -----')\n",
    "print (X.T) "
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "id": "HXIGyB5Fe4MQ"
   },
   "source": [
    "##  Addition and subtraction of matrices"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "id": "9og0LH4He4MQ"
   },
   "source": [
    "Addition and subtraction of matrices is performed element-wise, just as with vectors.  As with vectors two matrices must have the same dimensions in order to perform addition/subtraction on two matrices.  For example with two $P\\times N$ matrices\n",
    "\n",
    "$$\n",
    "\\mathbf{X}=\\begin{bmatrix}\n",
    "x_{11} & x_{12} & \\cdots & x_{1N}\\\\\n",
    "x_{21} & x_{22} & \\cdots & x_{2N}\\\\\n",
    "\\vdots & \\vdots & \\ddots & \\vdots\\\\\n",
    "x_{P1} & x_{P2} & \\cdots & x_{PN}\n",
    "\\end{bmatrix} \\,\\,\\,\\,\\,\\,\\,\\,\\,  \\mathbf{Y}=\\begin{bmatrix}\n",
    "y_{11} & y_{12} & \\cdots & y_{1N}\\\\\n",
    "y_{21} & y_{22} & \\cdots & y_{2N}\\\\\n",
    "\\vdots & \\vdots & \\ddots & \\vdots\\\\\n",
    "y_{P1} & y_{P2} & \\cdots & y_{PN}\n",
    "\\end{bmatrix}\n",
    "$$\n",
    "\n",
    "their element-wise sum is\n",
    "\n",
    "$$\n",
    "\\mathbf{X}+\\mathbf{Y}=\\begin{bmatrix}\n",
    "x_{11}+y_{11} & x_{12}+y_{12} & \\cdots & x_{1N}+y_{1N}\\\\\n",
    "x_{21}+y_{21} & x_{22}+y_{22} & \\cdots & x_{2N}+y_{2N}\\\\\n",
    "\\vdots & \\vdots & \\ddots & \\vdots\\\\\n",
    "x_{P1}+y_{P1} & x_{P2}+y_{P2} & \\cdots & x_{PN}+y_{PN}\n",
    "\\end{bmatrix}\n",
    "$$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "id": "rJt0w8K_e4MQ"
   },
   "source": [
    "Addition / subtraction of matrices using numpy is very done precisely as with vectors - with numpy both are referred to as *arrays*."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {
    "id": "4623b4zAe4MQ",
    "outputId": "4466f759-caac-4454-91cb-a7b1f4980bdd"
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "----- the matrix X -----\n",
      "[[1 3 1]\n",
      " [2 5 1]]\n",
      "----- the matrix Y -----\n",
      "[[ 5  9 14]\n",
      " [ 1  2  1]]\n",
      "----- the matrix X + Y -----\n",
      "[[ 6 12 15]\n",
      " [ 3  7  2]]\n"
     ]
    }
   ],
   "source": [
    "# create two matrices\n",
    "X = np.array([[1,3,1],[2,5,1]])\n",
    "Y = np.array([[5,9,14],[1,2,1]])\n",
    "print ('----- the matrix X -----')\n",
    "print (X) \n",
    "print ('----- the matrix Y -----')\n",
    "print (Y)\n",
    "\n",
    "# add  matrices\n",
    "print ('----- the matrix X + Y -----')\n",
    "print (X + Y)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "id": "s4-y8blme4MR"
   },
   "source": [
    "## Multplication"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "id": "-T8mh4uNe4MR"
   },
   "source": [
    "### Multiplication by a scalar"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "id": "vu063-fNe4MR"
   },
   "source": [
    "As with vectors we can multiply a matrix by a scalar - and this operation is performed element-by-element.  For any scalar value $c$ we write scalar multiplication as \n",
    "\n",
    "$$\n",
    "c\\times\\mathbf{X}=\\begin{bmatrix}\n",
    "c\\times x_{11} & c\\times x_{12} & \\cdots & c\\times x_{1N}\\\\\n",
    "c\\times x_{21} & c\\times x_{22} & \\cdots & c\\times x_{2N}\\\\\n",
    "\\vdots & \\vdots & \\ddots & \\vdots\\\\\n",
    "c\\times x_{P1} & c\\times x_{P2} & \\cdots & c\\times x_{PN}\n",
    "\\end{bmatrix}\n",
    "$$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "id": "mPLW6Ad9e4MR"
   },
   "source": [
    "In numpy scalar multiplication can be written very naturally using the '*' symbol, as illustrated in the next Python cell."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {
    "id": "uu7SsMSRe4MR",
    "outputId": "b00dadec-6bba-48f1-89c3-3f126ece87d7"
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "[[ 2  6  2]\n",
      " [ 4 10  2]]\n"
     ]
    }
   ],
   "source": [
    "# define a matrix\n",
    "X = np.array([[1,3,1],[2,5,1]])\n",
    "c = 2\n",
    "print (c*X)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "id": "qRVlkMure4MS"
   },
   "source": [
    "### Multiplication of a matrix by a vector"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "id": "wTzQ2M2me4MS"
   },
   "source": [
    "Generally speaking there are two ways to multiply an $P\\times N$ matrix $\\mathbf{X}$ by a vector $\\mathbf{a}$.  The first - referred to as *left multiplication* - involves multiplication by $1\\times P$ row vector $\\mathbf{a}$.  This operation is written $\\mathbf{a}\\mathbf{X} = \\mathbf{b}$, with $\\mathbf{b}$ being a $1\\times N$ dimensional vector.  It is defined by taking the inner product of $\\mathbf{a}$ with each column of $\\mathbf{X}$.\n",
    "\n",
    "$$\n",
    "\\mathbf{a}\\mathbf{X} = \\mathbf{b} = \n",
    "\\begin{bmatrix}\n",
    "\\sum_{p=1}^P a_px_{p1} \\,\\,\\,\\,\\,\n",
    "\\sum_{p=1}^P a_px_{p2} \\,\\,\\,\\,\\, \n",
    "\\cdots \\,\\,\\,\\,\\,\n",
    "\\sum_{p=1}^P a_px_{pN} \n",
    "\\end{bmatrix}\n",
    "$$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "id": "hxZAeFSCe4MS"
   },
   "source": [
    "Since this multiplication consists of a sequence of inner products, we can use the inner or dot product notation in numpy to compute a left multiplication as illustrated in the next cell."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {
    "id": "kYTQgpgle4MS",
    "outputId": "b2f922d2-84d4-4903-cfbf-4d7ee9764698"
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "[[3 8 2]]\n"
     ]
    }
   ],
   "source": [
    "# define a matrix\n",
    "X = np.array([[1,3,1],[2,5,1]])\n",
    "a = np.array([1,1])\n",
    "a.shape = (1,2)\n",
    "\n",
    "# compute a left multiplication\n",
    "print (np.dot(a,X))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "id": "J8HPyoWje4MS"
   },
   "source": [
    "Right multiplication is defined by multiplying $\\mathbf{X}$ on the right by a $N\\times 1$ vector $\\mathbf{a}$.  Right multiplication is written as $\\mathbf{X}\\mathbf{a} = \\mathbf{b}$ and $\\mathbf{b}$ will be a $P\\times 1$ vector.  The right product is defined as \n",
    "\n",
    "$$\n",
    "\\mathbf{X}\\mathbf{a} = \\mathbf{b} = \n",
    "\\begin{bmatrix}\n",
    "\\sum_{n=1}^N a_nx_{1n} \\\\\n",
    "\\sum_{n=1}^N a_nx_{2n} \\\\ \n",
    "\\vdots \\\\\n",
    "\\sum_{n=1}^N a_nx_{Pn} \n",
    "\\end{bmatrix}\n",
    "$$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "id": "yB_OcaWZe4MT"
   },
   "source": [
    "Since the right multiplication also consists of a sequence of inner products, we can use the inner or dot product notation in numpy to compute a right multiplication as illustrated in the next cell."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {
    "id": "SaSfL8que4MT",
    "outputId": "eddd771f-8585-48f4-8e8c-d5f4a8481e2c"
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "[[5]\n",
      " [8]]\n"
     ]
    }
   ],
   "source": [
    "# define a matrix\n",
    "X = np.array([[1,3,1],[2,5,1]])\n",
    "a = np.array([1,1,1])\n",
    "a.shape = (3,1)\n",
    "\n",
    "# compute a right multiplication\n",
    "print (np.dot(X,a))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "collapsed": true,
    "id": "KLIB0Ij4e4MT"
   },
   "source": [
    "### Element-wise multiplication of two matrices"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "id": "G4xewXhTe4MT"
   },
   "source": [
    "As with vectors, we can define element-wise multiplication on two matrices of the same size.  Multiplying two $P\\times N$ matrices $\\mathbf{x}$ and $\\mathbf{y}$ together gives"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "id": "oBaxU2Ine4MT"
   },
   "source": [
    "$$\n",
    "\\mathbf{X}\\times \\mathbf{Y}=\\begin{bmatrix}\n",
    "x_{11}\\times y_{11} & x_{12}\\times y_{12} & \\cdots & x_{1N}+y_{1N}\\\\\n",
    "x_{21}\\times y_{21} & x_{22}\\times y_{22} & \\cdots & x_{2N}+y_{2N}\\\\\n",
    "\\vdots & \\vdots & \\ddots & \\vdots\\\\\n",
    "x_{P1}\\times y_{P1} & x_{P2}\\times y_{P2} & \\cdots & x_{PN}+y_{PN}\n",
    "\\end{bmatrix}\n",
    "$$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "id": "ZOUGXkx_e4MT"
   },
   "source": [
    "This can be easily computed in numpy, as illustrated in the next Python cell for two small example matrices."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {
    "id": "eko1PW_te4MU",
    "outputId": "8a2a99ac-ce76-41cc-ec71-a5ca0355add1"
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "----- the matrix X -----\n",
      "[[1 3 1]\n",
      " [2 5 1]]\n",
      "----- the matrix Y -----\n",
      "[[ 5  9 14]\n",
      " [ 1  2  1]]\n",
      "----- the matrix X * Y -----\n",
      "[[ 5 27 14]\n",
      " [ 2 10  1]]\n"
     ]
    }
   ],
   "source": [
    "# create two matrices\n",
    "X = np.array([[1,3,1],[2,5,1]])\n",
    "Y = np.array([[5,9,14],[1,2,1]])\n",
    "print ('----- the matrix X -----')\n",
    "print (X) \n",
    "print ('----- the matrix Y -----')\n",
    "print (Y)\n",
    "\n",
    "# add  matrices\n",
    "print ('----- the matrix X * Y -----')\n",
    "print (X*Y)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "id": "080_HeRGe4MU"
   },
   "source": [
    "### General multiplication of two matrices"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "id": "KVzAOY6ee4MU"
   },
   "source": [
    "The regular product (or simply product) of two matrices $\\mathbf{X}$ and $\\mathbf{Y}$ can be defined based on the vector outer product operation, provided that the number of columns in $\\mathbf{X}$ matches the number of rows in $\\mathbf{Y}$. That is, we must have $\\mathbf{X}$ and $\\mathbf{Y}$ of sizes $P\\times N$ and $N \\times Q$ respectively, for the matrix product to be defined as \n",
    "\n",
    "$$\\mathbf{XY}= \\sum_{n=1}^N \\mathbf{x}_{n}\\mathbf{y}_{n}^{T}$$\n",
    "\n",
    "where $\\mathbf{x}_{n}$ is the $n^{th}$ column of $\\mathbf{X}$, and $\\mathbf{y}_{n}^{T}$ is the transpose of the $n^{th}$ column of $\\mathbf{Y}^{T}$ (or equivalently, the $n^{th}$ row of $\\mathbf{Y}$). Note that each summand above is an outer-product matrix of size $P \\times Q$, and so too is the final matrix $\\mathbf{XY}$.\n",
    "\n",
    "Matrix multplication can also be defined entry-wise, using vector inner-products, where the entry in the $p^{th}$ row and $q^{th}$ column of $\\mathbf{XY}$ can be found as the inner-product of (transpose of) the $p^{th}$ row in $\\mathbf{X}$ and the $q^{th}$ column in $\\mathbf{Y}$.\n",
    "\n",
    "$$\\left(\\mathbf{XY}\\right)_{p,q}= \\mathbf{x}_{p}^{T}\\mathbf{y}_{q}$$"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {
    "id": "fUlhMYt7e4MU",
    "outputId": "202437ed-bc81-4010-9324-462aedf65afb"
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "----- the matrix X -----\n",
      "[[1 3 1]\n",
      " [2 5 1]]\n",
      "----- the matrix Y -----\n",
      "[[ 5  1]\n",
      " [ 9  2]\n",
      " [14  1]]\n",
      "----- the matrix XY -----\n",
      "[[46  8]\n",
      " [69 13]]\n"
     ]
    }
   ],
   "source": [
    "# create two matrices\n",
    "X = np.array([[1,3,1],[2,5,1]])\n",
    "Y = np.array([[5,9,14],[1,2,1]])\n",
    "Y = np.array([[5,1],[9,2],[14,1]])\n",
    "print ('----- the matrix X -----')\n",
    "print (X) \n",
    "print ('----- the matrix Y -----')\n",
    "print (Y)\n",
    "\n",
    "# add  matrices\n",
    "print ('----- the matrix XY -----')\n",
    "print (np.dot(X,Y))"
   ]
  }
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